5.10. Doppelfolgen und Doppelreihen


Wir erweitern in diesem Abschnitt den Folgenbegriff: Bisher haben wir uns mit eindimensional indizierten Listen reeller Zahlen befasst, eine Deutung, die auch durch unsere Notation unterstützt wird. Es liegt nahe, auch zweidimensionale Listen in unsere Überlegungen einzubeziehen. Zwar gewinnen wir dadurch i.w. keine neuen Inhalte (siehe etwa [5.10.9] - [5.10.12]), dennoch spielen Doppelfolgen unter technischen Gesichtspunkten eine wichtige Rolle. Mit ihrer Hilfe nämlich finden wir grundlegende Kriterien ([5.10.23] und [5.10.26]), die das Vertauschen von Grenzprozessen gestatten.

Definition:  Jede Funktion

( a nm ): × MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcacaGG6aGaeSyfHu6aaWbaaSqabeaacqGHxiIkaaGccqGHxdaTcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiabgkziUkabl2riHcaa@46A4@
[5.10.1]

nennen wir eine Doppelfolge (in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@36D9@ ).

( i=0 n j=0 m a ij ):× MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcacaGG6aGaeSyfHuQaey41aqRaeSyfHuQaeyOKH4QaeSyhHekaaa@500B@
[5.10.2]

heißt die zu  ( a nm ) n0 m0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcadaWgaaWcbaqbaeaabiqaaaqaaiaad6gacqGHLjYScaaIWaaabaGaamyBaiabgwMiZkaaicdaaaaabeaaaaa@426B@ gehörige Doppelreihe.

Beachte:

  • Wie bei den gewöhnlichen Folgen auch, werden wir Funktionen von k × l MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHi6aaWbaaSqabeaacqGHLjYScaWGRbaaaOGaey41aqRaeSijHi6aaWbaaSqabeaacqGHLjYScaWGSbaaaOGaeyOKH4QaeSyhHekaaa@43A8@ ebenfalls als Doppelfolgen bezeichnen. Wir notieren sie dann (wie in [5.10.2] bereits geschehen) in der Form  ( a nm ) nk ml MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcadaWgaaWcbaqbaeaabiqaaaqaaiaad6gacqGHLjYScaWGRbaabaGaamyBaiabgwMiZkaadYgaaaaabeaaaaa@42D8@ .

  • Wertetabellen von Doppelfolgen müssen natürlich flächenhaft angelegt werden. Dabei fassen wir n als nach unten laufenden Zeilenindex und m als nach rechts laufenden Spaltenindex auf. Z.B.:

    (n+ (1) n m)=( 0 1 2 3 3 4 5 6 2 1 0 1 5 6 7 8 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gacqGHRaWkcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccaWGTbGaaiykaiabg2da9iaacIcafaqabeqbfaaaaaqaaiaaicdaaeaacqGHsislcaaIXaaabaGaeyOeI0IaaGOmaaqaaiabgkHiTiaaiodaaeaacqWIVlctaeaacaaIZaaabaGaaGinaaqaaiaaiwdaaeaacaaI2aaabaGaeS47IWeabaGaaGOmaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGymaaqaaiabl+UimbqaaiaaiwdaaeaacaaI2aaabaGaaG4naaqaaiaaiIdaaeaacqWIVlctaeaacqWIUlstaeaacqWIUlstaeaacqWIUlstaeaacqWIUlstaeaacqWIXlYtaaGaaiykaaaa@612B@

     

Wir übertragen nun die bei den gewöhnlichen Folgen eingeführten Standardbegriffe.

Definition:  Eine Doppelfolge ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ heißt

1. beschränkt, falls es ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ und ein c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiolabl2riHcaa@3945@ gibt, so dass
 
| a nm |c  für alle  n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacYhacqGHKjYOcaWGJbGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGUbGaaiilaiaad2gacqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaaaa@4C70@

 
[5.10.3]
2. monoton wachsend, falls für alle ( n 1 , m 1 ),( n 2 , m 2 ) × MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaaiikaiaad6gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIYaaabeaakiaacMcacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiabgEna0kablwriLoaaCaaaleqabaGaey4fIOcaaaaa@4A70@ gilt:
 
n 1 n 2        m 1 m 2 a n 1 m 1 a n 2 m 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabgsMiJkaad6gadaWgaaWcbaGaaGOmaaqabaGccaaMe8Uaey4jIKTaaGjbVlaad2gadaWgaaWcbaGaaGymaaqabaGccqGHKjYOcaWGTbWaaSbaaSqaaiaaikdaaeqaaOGaaGzbVlabgkDiElaaywW7caWGHbWaaSbaaSqaaiaad6gadaWgaaadbaGaaGymaaqabaWccaaMc8UaamyBamaaBaaameaacaaIXaaabeaaaSqabaGccqGHKjYOcaWGHbWaaSbaaSqaaiaad6gadaWgaaadbaGaaGOmaaqabaWccaaMc8UaamyBamaaBaaameaacaaIYaaabeaaaSqabaaaaa@5935@

 
[5.10.4]
3. monoton fallend, falls für alle ( n 1 , m 1 ),( n 2 , m 2 ) × MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaaiikaiaad6gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIYaaabeaakiaacMcacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiabgEna0kablwriLoaaCaaaleqabaGaey4fIOcaaaaa@4A70@ gilt:
 
n 1 n 2        m 1 m 2 a n 1 m 1 a n 2 m 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabgsMiJkaad6gadaWgaaWcbaGaaGOmaaqabaGccaaMe8Uaey4jIKTaaGjbVlaad2gadaWgaaWcbaGaaGymaaqabaGccqGHKjYOcaWGTbWaaSbaaSqaaiaaikdaaeqaaOGaaGzbVlabgkDiElaaywW7caWGHbWaaSbaaSqaaiaad6gadaWgaaadbaGaaGymaaqabaWccaaMc8UaamyBamaaBaaameaacaaIXaaabeaaaSqabaGccqGHLjYScaWGHbWaaSbaaSqaaiaad6gadaWgaaadbaGaaGOmaaqabaWccaaMc8UaamyBamaaBaaameaacaaIYaaabeaaaSqabaaaaa@5946@

 
[5.10.5]
4. konvergent gegen g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgIGiolabl2riHcaa@3949@   (in Zeichen: a nm g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4Qaam4zaaaa@3CCE@ ), falls es zu jedem ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@
   ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ gibt, so dass
 
| a nm g|<ε  für alle  n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiabgkHiTiaadEgacaGG8bGaeyipaWJaeqyTduMaaeOzaiaabYpacaqGYbGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacaGGSaGaamyBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@4DB4@

 
[5.10.6]
5. Cauchy-Folge, falls es zu jedem ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ gibt, so dass
 
| a n 1 m 1 a n 2 m 2 |<ε  für alle   n 1 > n 2 n 0        m 1 > m 2 n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIXaaabeaaliaaykW7caWGTbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiabgkHiTiaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIYaaabeaaliaaykW7caWGTbWaaSbaaWqaaiaaikdaaeqaaaWcbeaakiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gadaWgaaWcbaGaaGymaaqabaGccqGH+aGpcaWGUbWaaSbaaSqaaiaaikdaaeqaaOGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaakiaaysW7cqGHNis2caaMe8UaamyBamaaBaaaleaacaaIXaaabeaakiabg6da+iaad2gadaWgaaWcbaGaaGOmaaqabaGccqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaaaa@6541@
[5.10.7]

Beachte:

  • Die Formulierung der Beschränktheit (und auch die der Cauchy-Bedingung) ist der Natur der Doppelfolgen angepasst und keine direkte Übertragung der Originaldefinition. Dennoch ist sie eine - und wie die weiteren Ausführungen zeigen - sinnvolle Verallgemeinerung von [5.3.11], denn man hat ja

    | a n |c  für alle  n n 0 | a n |max{c,| a 1 |,,| a n 0 1 |}  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaaqabaGccaGG8bGaeyizImQaam4yaiaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaGccaaMf8UaeyO0H4TaaGzbVlaacYhacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaaiiFaiabgsMiJkGac2gacaGGHbGaaiiEaiaacUhacaWGJbGaaiilaiaacYhacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiiFaiaacYcacqWIMaYscaGGSaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIWaaabeaaliabgkHiTiaaigdaaeqaaOGaaiiFaiaac2hacaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gaaaa@6FA3@

     

Beispiel:  

  • (1) n+m ( 1 n + 1 m )0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGUbGaey4kaSIaamyBaaaakiaacIcadaWcaaqaaiaaigdaaeaacaWGUbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaad2gaaaGaaiykaiabgkziUkaaicdaaaa@43C5@
[5.10.8]

Denn wählt man zu ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ ein n 0 > 2 ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabg6da+maalaaabaGaaGOmaaqaaiabew7aLbaaaaa@3AC7@ , so gilt für alle n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@3B9D@ :

| (1) n+m ( 1 n + 1 m )|= 1 n + 1 m < ε 2 + ε 2 =ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaiabgUcaRiaad2gaaaGccaGGOaWaaSaaaeaacaaIXaaabaGaamOBaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGTbaaaiaacMcacaGG8bGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGTbaaaiabgYda8maalaaabaGaeqyTdugabaGaaGOmaaaacqGHRaWkdaWcaaqaaiabew7aLbqaaiaaikdaaaGaeyypa0JaeqyTdugaaa@51FA@

Die neuen Begriffe sind den alten so parallel angelegt, dass man auch gleiche Eigenschaften erwarten darf. Die folgende Bemerkung erleichtert deren Nachweis. Zu ihrer Formulierung vereinbaren wir: Eine Funktion

ϕ=( ϕ 1 , ϕ 2 ): × MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaeyypa0Jaaiikaiabew9aQnaaBaaaleaacaaIXaaabeaakiaacYcacqaHvpGAdaWgaaWcbaGaaGOmaaqabaGccaGGPaGaaiOoaiablwriLoaaCaaaleqabaGaey4fIOcaaOGaeyOKH4QaeSyfHu6aaWbaaSqabeaacqGHxiIkaaGccqGHxdaTcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@4C2D@

heißt streng wachsend falls die Koordinatenfunktionen ϕ 1  und  ϕ 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaaigdaaeqaaOGaaeyDaiaab6gacaqGKbGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaaaa@3DAA@ streng monoton wachsend sind.

Man beachte, dass für jede Doppelfolge ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ die Funktion ( a nm )ϕ=( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcacqWIyiYBcqaHvpGAcqGH9aqpcaGGOaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaaiykaaaa@45E7@ eine gewöhnliche Folge ist. Ferner zeigt man leicht per Induktion:

ϕ i (n)n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad6gacaGGPaGaeyyzImRaamOBaaaa@3D5E@   für alle n

 

Bemerkung:  Es sei ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ irgendeine Doppelfolge, dann gilt:

1. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ beschränkt ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caGGOaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaaiykaaaa@416E@ beschränkt für alle streng wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ [5.10.9]
2. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ monoton ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caGGOaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaaiykaaaa@416E@ monoton für alle streng wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ [5.10.10]
3. a nm g a ϕ(n) g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4Qaam4zaiaaywW7cqGHuhY2caaMf8UaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaeyOKH4Qaam4zaaaa@4A53@   für alle streng wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ [5.10.11]
4. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ Cauchy-Folge ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caGGOaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaaiykaaaa@416E@ Cauchy-Folge für alle streng
wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@
[5.10.12]

Beweis:  
1.  

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ "  Sei | a nm |c  für alle  n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacYhacqGHKjYOcaWGJbGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGUbGaaiilaiaad2gacqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaaaa@4C70@ .

Für n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@39FB@   ist  ϕ i (n) ϕ i ( n 0 ) n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad6gacaGGPaGaeyyzImRaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad6gadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@4636@ , also hat man: | a ϕ(n) |c  für alle  n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacYhacqGHKjYOcaWGJbGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGUbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@4B76@ , und damit:

| a ϕ(n) |max{c,| a ϕ(1) |,,| a ϕ( n 0 1) |}  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacYhacqGHKjYOciGGTbGaaiyyaiaacIhacaGG7bGaam4yaiaacYcacaGG8bGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaaGymaiaacMcaaeqaaOGaaiiFaiaacYcacqWIMaYscaGGSaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gadaWgaaadbaGaaGimaaqabaWccqGHsislcaaIXaGaaiykaaqabaGccaGG8bGaaiyFaiaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamOBaaaa@60A7@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ "  zeigen wir indirekt: Ist ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ unbeschränkt, so ist insbesondere die Aussage

| a nm |k  für alle  n,mk+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacYhacqGHKjYOcaWGRbGaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGUbGaaiilaiaad2gacqGHLjYScaWGRbGaey4kaSIaaGymaaaa@4D2C@

für kein k gültig. Also gibt es zu jedem  k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A65@ Zahlen n k , m k >k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbaabeaakiaacYcacaWGTbWaaSbaaSqaaiaadUgaaeqaaOGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGH+aGpcaWGRbaaaaaa@3F5F@ , so dass

| a n k    m k |>k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaWGRbaabeaaliaaykW7caWGTbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiaacYhacqGH+aGpcaWGRbaaaa@403D@ .

Wir setzen nun rekursiv

ϕ(1)( n 1 , m 1 ) ϕ(i+1)( n p , m p ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabew9aQjaacIcacaaIXaGaaiykaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaaGymaaqabaGccaGGPaaabaGaeqy1dOMaaiikaiaadMgacqGHRaWkcaaIXaGaaiykaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaadchaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaamiCaaqabaGccaGGPaaaaaaa@4D25@

wobei  pmax{ ϕ 1 (1), ϕ 2 (1),, ϕ 1 (i), ϕ 2 (i),i+1} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iGac2gacaGGHbGaaiiEaiaacUhacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGymaiaacMcacaGGSaGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacaGGPaGaaiilaiablAciljaacYcacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaamyAaiaacMcacaGGSaGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadMgacaGGPaGaaiilaiaadMgacqGHRaWkcaaIXaGaaiyFaaaa@5701@ .

ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ ist streng wachsend, denn für alle i,j ,   ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaacYcacaWGQbGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGHxiIkaaGccaGGSaGaaGjbVlaadMgacqGHLjYScaWGQbaaaa@41EC@ , hat man:

ϕ 1 (i+1)= n p >p ϕ 1 (j) ϕ 2 (i+1)= m p >p ϕ 2 (j) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabew9aQnaaBaaaleaacaaIXaaabeaakiaacIcacaWGPbGaey4kaSIaaGymaiaacMcacqGH9aqpcaWGUbWaaSbaaSqaaiaadchaaeqaaOGaeyOpa4JaamiCaiabgwMiZkabew9aQnaaBaaaleaacaaIXaaabeaakiaacIcacaWGQbGaaiykaaqaaiabew9aQnaaBaaaleaacaaIYaaabeaakiaacIcacaWGPbGaey4kaSIaaGymaiaacMcacqGH9aqpcaWGTbWaaSbaaSqaaiaadchaaeqaaOGaeyOpa4JaamiCaiabgwMiZkabew9aQnaaBaaaleaacaaIYaaabeaakiaacIcacaWGQbGaaiykaaaaaaa@5A90@

Die Abschätzung | a ϕ(i+1) |=| a n p    m p |>pi+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaadMgacqGHRaWkcaaIXaGaaiykaaqabaGccaGG8bGaeyypa0JaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaWGWbaabeaaliaaykW7caWGTbWaaSbaaWqaaiaadchaaeqaaaWcbeaakiaacYhacqGH+aGpcaWGWbGaeyyzImRaamyAaiabgUcaRiaaigdaaaa@4E6F@ zeigt nun, dass ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ unbeschränkt ist.   Widerspruch!
 

2.  

Wir betrachten beispielhaft nur monoton wachsende Folgen.

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ":  Für ji MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgsMiJkaadMgaaaa@38FB@ hat man  ϕ 1 (j) ϕ 1 (i)       ϕ 2 (j) ϕ 2 (i) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadQgacaGGPaGaeyizImQaeqy1dO2aaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadMgacaGGPaGaaGjbVlabgEIizlaaysW7cqaHvpGAdaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamOAaiaacMcacqGHKjYOcqaHvpGAdaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamyAaiaacMcaaaa@51AF@   und damit: a ϕ(j) a ϕ(i) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOAaiaacMcaaeqaaOGaeyizImQaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamyAaiaacMcaaeqaaaaa@4173@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Auch hier argumentieren wir indirekt. Angenommen es gibt zwei Zahlenpaare ( n 1 , m 1 ),( n 2 , m 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaaiikaiaad6gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIYaaabeaakiaacMcaaaa@41BB@   mit  n 1 < n 2        m 1 < m 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabgYda8iaad6gadaWgaaWcbaGaaGOmaaqabaGccaaMe8Uaey4jIKTaaGjbVlaad2gadaWgaaWcbaGaaGymaaqabaGccqGH8aapcaWGTbWaaSbaaSqaaiaaikdaaeqaaaaa@43BF@ , so dass a n 1 m 1 > a n 2 m 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbWaaSbaaWqaaiaaigdaaeqaaSGaaGPaVlaad2gadaWgaaadbaGaaGymaaqabaaaleqaaOGaeyOpa4JaamyyamaaBaaaleaacaWGUbWaaSbaaWqaaiaaikdaaeqaaSGaaGPaVlaad2gadaWgaaadbaGaaGOmaaqabaaaleqaaaaa@434D@ . Wir setzen nun

ϕ(1)( n 1 , m 1 ),ϕ(2)( n 2 , m 2 )undϕ(i+2)( n 2 +i, m 2 +i) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaaiikaiaaigdacaGGPaGaeyypa0Jaaiikaiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaaGzbVlabew9aQjaacIcacaaIYaGaaiykaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaaGzbVlaabwhacaqGUbGaaeizaiaaywW7cqaHvpGAcaGGOaGaamyAaiabgUcaRiaaikdacaGGPaGaeyypa0Jaaiikaiaad6gadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGPbGaaiilaiaad2gadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGPbGaaiykaaaa@632B@ .

ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ ist offensichtlich streng wachsend, denn

( ϕ 1 )=( n 1 , n 2 , n 2 +1, n 2 +2,) ( ϕ 2 )=( m 1 , m 2 , m 2 +1, m 2 +2,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5FBC@

aber ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ wächst nicht monoton.   Widerspruch!
 

3.  

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ":  Zu ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ gibt es ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ , so dass

| a nm g|<ε  für alle  n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiabgkHiTiaadEgacaGG8bGaeyipaWJaeqyTduMaaeOzaiaabYpacaqGYbGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacaGGSaGaamyBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@4DB4@ .

Da ϕ i (n)n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad6gacaGGPaGaeyyzImRaamOBaaaa@3D5E@ hat man somit für alle n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@39FB@ erst recht:

| a ϕ(n) g|<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiabgkHiTiaadEgacaGG8bGaeyipaWJaeqyTdugaaa@4121@

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Wir gehen noch einmal indirekt vor und nehmen an, ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ konvergiere nicht gegen g. Dann gibt es ein ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ , derart dass zu jedem k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A65@ Zahlen n k , m k k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbaabeaakiaacYcacaWGTbWaaSbaaSqaaiaadUgaaeqaaOGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGHLjYScaWGRbGaey4kaSIaaGymaaaaaaa@41BA@ gibt mit

| a n k m k g|ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaWGRbaabeaaliaaykW7caWGTbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiabgkHiTiaadEgacaGG8bGaeyyzImRaeqyTdugaaa@438B@

Ähnlich wie in 1. liefert nun die Rekursion

ϕ(1)( n 1 , m 1 ) ϕ(i+1)( n p , m p ) , MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabew9aQjaacIcacaaIXaGaaiykaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaaGymaaqabaGccaGGPaaabaGaeqy1dOMaaiikaiaadMgacqGHRaWkcaaIXaGaaiykaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaadchaaeqaaOGaaiilaiaad2gadaWgaaWcbaGaamiCaaqabaGccaGGPaaaaaaa@4D25@

mit  pmax{ ϕ 1 (1), ϕ 2 (1),, ϕ 1 (i), ϕ 2 (i)} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iGac2gacaGGHbGaaiiEaiaacUhacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGymaiaacMcacaGGSaGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacaGGPaGaaiilaiablAciljaacYcacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaamyAaiaacMcacaGGSaGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadMgacaGGPaGaaiyFaaaa@53C6@ , ein streng wachsendes ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ , so dass

| a ϕ(n) g|ε  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiabgkHiTiaadEgacaGG8bGaeyyzImRaeqyTduMaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaWGUbaaaa@4A80@ .

( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ ist damit divergent.   Widerspruch!
 

4.  

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ":  Hat man für ein beliebiges ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@

| a n 1 m 1 a n 2 m 2 |<ε  für alle   n 1 > n 2 n 0        m 1 > m 2 n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIXaaabeaaliaaykW7caWGTbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiabgkHiTiaadggadaWgaaWcbaGaamOBamaaBaaameaacaaIYaaabeaaliaaykW7caWGTbWaaSbaaWqaaiaaikdaaeqaaaWcbeaakiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gadaWgaaWcbaGaaGymaaqabaGccqGH+aGpcaWGUbWaaSbaaSqaaiaaikdaaeqaaOGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaakiaaysW7cqGHNis2caaMe8UaamyBamaaBaaaleaacaaIXaaabeaakiabg6da+iaad2gadaWgaaWcbaGaaGOmaaqabaGccqGHLjYScaWGUbWaaSbaaSqaaiaaicdaaeqaaaaa@6541@ ,

so gilt insbesondere (beachte: ϕ i (n)> ϕ i (m)m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad6gacaGGPaGaeyOpa4Jaeqy1dO2aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaad2gacaGGPaGaeyyzImRaamyBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@473F@ )

| a ϕ(n) a ϕ(m) |<ε  für alle  n>m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiabgkHiTiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad2gacaGGPaaabeaakiaacYhacqGH8aapcqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gacqGH+aGpcaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@539E@ .

Die Ungleichung ist auch für nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgsMiJkaad2gaaaa@3903@ gültig (Vertauschen der Summandenreihenfolge), also für alle n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@3B9D@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Ist ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ keine Cauchy-Folge, so gibt es ein ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ , derart dass es zu jedem k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A65@ Zahlen  n k > r k >k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+iaadkhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpcaWGRbaaaa@3C9F@ und m k > s k >k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGRbaabeaakiabg6da+iaadohadaWgaaWcbaGaam4AaaqabaGccqGH+aGpcaWGRbaaaa@3C9F@ gibt mit

| a n k m k a r k s k |ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBamaaBaaameaacaWGRbaabeaaliaaykW7caWGTbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiabgkHiTiaadggadaWgaaWcbaGaamOCamaaBaaameaacaWGRbaabeaaliaaykW7caWGZbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiaacYhacqGHLjYScqaH1oqzaaa@4985@ .

Durch die zweistufige Rekursion

ϕ(1)( r 1 , s 1 )      ϕ(2)( n 1 , m 1 ) ϕ(2i+1)( r p , s p )      ϕ(2i+2)( n p , m p ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FF3@

mit  pmax{ ϕ 1 (1), ϕ 2 (1),, ϕ 1 (2i), ϕ 2 (2i)} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iGac2gacaGGHbGaaiiEaiaacUhacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGymaiaacMcacaGGSaGaeqy1dO2aaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacaGGPaGaaiilaiablAciljaacYcacqaHvpGAdaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGOmaiaadMgacaGGPaGaaiilaiabew9aQnaaBaaaleaacaaIYaaabeaakiaacIcacaaIYaGaamyAaiaacMcacaGG9baaaa@553E@ wird ein streng wachsendes ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ erzeugt, so dass

| a ϕ(2n) a ϕ(2n1) |ε  für alle  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaaikdacaWGUbGaaiykaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiabew9aQjaacIcacaaIYaGaamOBaiabgkHiTiaaigdacaGGPaaabeaakiaacYhacqGHLjYScqaH1oqzcaqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaad6gaaaa@51E8@ .

( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ kann also keine Cauchy-Folge sein.   Widerspruch!

[5.10.11] garantiert nun auch bei Doppelfolgen die Eindeutigkeit des Grenzwerts: Hätte nämlich ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ zwei verschiedene Grenzwerte, so z.B. auch die gewöhnliche Folge ( a nn ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGUbaabeaakiaacMcaaaa@3B4F@ . Die Konvergenz a nm g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4Qaam4zaaaa@3CCE@ dürfen wir jetzt also auch so notieren:

lim n,m a nm =g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacaGGSaGaamyBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaGccqGH9aqpcaWGNbaaaa@44ED@
 

Viele Eigenschaften gewöhnlicher Folgen lassen sich direkt aus [5.10.9] bis [5.10.12] ableiten. So etwa die zentralen Grenzwertsätze:

Bemerkung:  

1. a nm a       b nm b a nm + b nm a+b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiaaysW7cqGHNis2caaMe8UaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamOyaiaaywW7cqGHshI3caaMf8UaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiabgUcaRiaadkgaaaa@5D01@ [5.10.13]
2. a nm a       b nm b a nm b nm ab MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiaaysW7cqGHNis2caaMe8UaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamOyaiaaywW7cqGHshI3caaMf8UaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOeI0IaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiabgkHiTiaadkgaaaa@5D17@ [5.10.14]
3. a nm a       b nm b a nm b nm ab MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiaaysW7cqGHNis2caaMe8UaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamOyaiaaywW7cqGHshI3caaMf8UaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyyXICTaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiabgwSixlaadkgaaaa@5FD1@ [5.10.15]
4. a nm a       b nm b a nm b nm a b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiaaysW7cqGHNis2caaMe8UaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamOyaiaaywW7cqGHshI3caaMf8+aaSaaaeaacaWGHbWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaaaaOGaeyOKH46aaSaaaeaacaWGHbaabaGaamOyaaaaaaa@5B5D@  ,   falls    b, b nm 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacYcacaWGIbWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaGccqGHGjsUcaaIWaaaaa@3E0E@ [5.10.16]

Beweis:  Wir führen beispielhaft nur den Beweis zu 1.: Man hat für jedes streng wachsende ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ :

a ϕ(n) a       b ϕ(n) b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaeyOKH4QaamyyaiaaysW7cqGHNis2caaMe8UaamOyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaeyOKH4QaamOyaaaa@4A41@

Also folgt mit dem ersten Grenzwertsatz (siehe [5.6.1]):  a ϕ(n) + b ϕ(n) a+b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaey4kaSIaamOyamaaBaaaleaacqaHvpGAcaGGOaGaamOBaiaacMcaaeqaaOGaeyOKH4QaamyyaiabgUcaRiaadkgaaaa@4550@ , d.h. aber:

a nm + b nm a+b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4QaamyyaiabgUcaRiaadkgaaaa@4400@ .

Aber auch wichtige Konvergenzkriterien können nun mühelos auf Doppelfolgen übertragen werden.

Bemerkung:  

1. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ konvergent ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caGGOaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaaiykaaaa@40C7@ beschränkt [5.10.17]
2. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ monoton und beschränkt ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caGGOaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaaiykaaaa@40C7@ konvergent [5.10.18]
3. ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ konvergent ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caGGOaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaaiykaaaa@40C6@ Cauchyfolge [5.10.19]

Beweis:  Auch hier beweisen wir exemplarisch nur die erste Aussage:

( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ konvergent
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ konvergent für alle streng wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@  [5.10.11]
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ( a ϕ(n) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaeqy1dOMaaiikaiaad6gacaGGPaaabeaakiaacMcaaaa@3BF6@ beschränkt für alle streng wachsenden ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@  [5.5.1]
MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ beschränkt  [5.10.9]

Die zweidimensionale Struktur der Doppelfolgen erlaubt es, einen weiteren Konvergenzbegriff einzuführen. Mit seiner Hilfe wird die Ermittlung des Limes gemäß [5.10.6] in manchen Fällen erleichtert.

Definition:  Wir nennen eine Doppelfolge ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@   zeilenkonvergent, wenn für jedes feste n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3A68@ die gewöhnliche Folge ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ konvergent ist. Die Zahlen

lim m a nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad2gacqGHsgIRcqGHEisPaeqaaOGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaaaa@414E@
[5.10.20]

heißen Zeilengrenzwerte. Analog richten wir die Begriffe spaltenkonvergent und Spaltengrenzwert ein.

Beachte:

  • Der neue und der alte Konvergenzbegriff beschreiben nicht diesselben Verhältnisse:

    Die Doppelfolge ( (1) n+m ( 1 n + 1 m )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaiabgUcaRiaad2gaaaGccaGGOaWaaSaaaeaacaaIXaaabaGaamOBaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGTbaaaiaacMcacaGGPaaaaa@4277@ etwa konvergiert nach Beispiel [5.10.8] gegen 0, sie ist aber weder zeilen- noch spaltenkonvergent. Für (sogar) jedes feste n nämlich belegen die Konvergenzen

    (1) n+2m ( 1 n + 1 2m )= (1) n n + (1) n 2m (1) n n (1) n+2m+1 ( 1 n + 1 2m+1 )= (1) n n (1) n 2m+1 (1) n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7EBF@

    dass die Folge ( (1) n+m ( 1 n + 1 m )) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaiabgUcaRiaad2gaaaGccaGGOaWaaSaaaeaacaaIXaaabaGaamOBaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGTbaaaiaacMcacaGGPaaaaa@4277@ zwei verschiedene Häufungspunkte besitzt und somit divergent sein muss. Dies belegt die Zeilendivergenz. Analog zeigt man die Spaltendivergenz.

    Die Doppelfolge ( (1 1 n ) m ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOBaaaacaGGPaWaaWbaaSqabeaacaWGTbaaaOGaaiykaaaa@3CAA@ ist zeilenkonvergent (  (1 1 n ) m 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad2gaaaGccqGHsgIRcaaIWaaaaa@3DF8@  ) und spaltenkonvergent (  (1 1 n ) m 1 m =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaacaWGUbaaaiaacMcadaahaaWcbeqaaiaad2gaaaGccqGHsgIRcaaIXaWaaWbaaSqabeaacaWGTbaaaOGaeyypa0JaaGymaaaa@40E3@  ) aber, wie die nächste Bemerkung zeigt, nicht konvergent.
     

Obwohl - wie gerade gesehen - die beiden Konvergenzbegriffe unverträglich sind, spielen konvergente Doppelfolgen, die gleichzeitig zeilen- und spaltenkonvergent sind, eine wichtige Rolle. Bei ihnen nämlich darf man die Reihenfolge der Grenzprozesse vertauschen. Man beachte, dass bei der Formulierung der nachfolgenden Bemerkung die Unterscheidung zwischen nicht festen und festen Indizes nicht mehr durch die Verwendung kursiver bzw. nicht nicht-kursiver Schrift unterstützt werden kann.

Bemerkung:   ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ sei eine gegen g konvergente Doppelfolge. Dann gilt:

  1. Ist ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ zeilenkonvergent, so konvergieren die Zeilengrenzwerte gegen g:

    lim n ( lim m a nm )=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGaaiikamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWGTbGaeyOKH4QaeyOhIukabeaakiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcacqGH9aqpcaWGNbaaaa@4C07@

     
    [5.10.21]
  2. Ist ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ spaltenkonvergent, so konvergieren die Spaltengrenzwerte gegen g:

    lim m ( lim n a nm )=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad2gacqGHsgIRcqGHEisPaeqaaOGaaiikamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcacqGH9aqpcaWGNbaaaa@4C07@

     
    [5.10.22]
  3. Ist ( a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcaaaa@3B4E@ zeilen- und spaltenkonvergent, so ist die doppelte Grenzwertbildung möglich und von der Reihenfolge unabhängig:

    lim n ( lim m a nm )= lim m ( lim n a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGaaiikamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWGTbGaeyOKH4QaeyOhIukabeaakiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiaacMcacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamyBaiabgkziUkabg6HiLcqabaGccaGGOaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaaiykaaaa@5FC7@
    [5.10.23]

Beweis:  Wir zeigen nur 1., denn 2. beweist man analog und 3. ist eine direkte Folgerung aus 1. und 2.

1.  

Wir setzen zur Abkürzung  g n lim m a nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBaaaleaacaWGUbaabeaakiabg2da9maaxababaGaciiBaiaacMgacaGGTbaaleaacaWGTbGaeyOKH4QaeyOhIukabeaakiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaaaaa@4469@   und zeigen: g n g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBaaaleaacaWGUbaabeaakiabgkziUkaadEgaaaa@3A57@ . Sei dazu ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@38D2@ vorgegeben. Da  a nm g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbGaaGPaVlaad2gaaeqaaOGaeyOKH4Qaam4zaaaa@3CCE@ , gibt es ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaakiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3B58@ , so dass für alle n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@39FB@ gilt:

| a nm g|< ε 2   für alle  m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamOBaiaaykW7caWGTbaabeaakiabgkHiTiaadEgacaGG8bGaeyipaWZaaSaaaeaacqaH1oqzaeaacaaIYaaaaiaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamyBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@4D80@

d.h. aber für diese m g ε 2 < a nm <g+ ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgkHiTmaalaaabaGaeqyTdugabaGaaGOmaaaacqGH8aapcaWGHbWaaSbaaSqaaiaad6gacaaMc8UaamyBaaqabaGccqGH8aapcaWGNbGaey4kaSYaaSaaaeaacqaH1oqzaeaacaaIYaaaaaaa@448A@ und damit für den Grenzwert g n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBaaaleaacaWGUbaabeaaaaa@3774@ :

g ε 2 g n g+ ε 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgkHiTmaalaaabaGaeqyTdugabaGaaGOmaaaacqGHKjYOcaWGNbWaaSbaaSqaaiaad6gaaeqaaOGaeyizImQaam4zaiabgUcaRmaalaaabaGaeqyTdugabaGaaGOmaaaaaaa@4375@

Also hat man:  | g n g| ε 2 <ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadEgadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGNbGaaiiFaiabgsMiJoaalaaabaGaeqyTdugabaGaaGOmaaaacqGH8aapcqaH1oqzaaa@422A@ .

Die bisher erzielten Ergebnisse gelten natürlich insbesondere auch für Doppelreihen. Uns interessiert hier vor allem die Aussage [5.10.23], die wir im Zusammenhang mit absolut konvergenten Doppelreihen neu formulieren wollen. Zunächst betrachten wir dazu Doppelreihen mit positiven Gliedern.

Bemerkung:  Eine Doppelreihe ( i=0 n j=0 m a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4701@ mit positiven Gliedern ist genau dann konvergent, wenn sie beschränkt ist. In diesem Fall gilt die folgende Abschätzung:

i=0 n j=0 m a ij i=0 j=0 a ij   für alle  n,m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaadaaeWbqaaiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaaaeaacaWGQbGaeyypa0JaaGimaaqaaiaad2gaa0GaeyyeIuoaaSqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aOGaeyizIm6aaabCaeaadaaeWbqaaiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaaaeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaWcbaGaamyAaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakiaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaamOBaiaacYcacaWGTbGaeyicI4SaeSyfHukaaa@65C8@
[5.10.24]

Beweis:   Die Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ " steht in [5.10.17].

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Da alle a ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaaaa@39E3@ positiv sind, wächst die Doppelreihe monoton. Die Konvergenz folgt also aus [5.10.18].

Nach [5.10.11] kann man für jedes streng wachsende ϕ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOgaaa@3735@ den Grenzwert gemäß Beweis zu [5.7.1] auch als Supremum der monotonen wachsenden ([5.10.10]) gewöhnlichen Folge ( i=0 ϕ 1 (n) j=0 ϕ 2 (n) a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacqaHvpGAdaWgaaadbaGaaGOmaaqabaWccaGGOaGaamOBaiaacMcaa0GaeyyeIuoaaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeqy1dO2aaSbaaWqaaiaaigdaaeqaaSGaaiikaiaad6gacaGGPaaaniabggHiLdGccaGGPaaaaa@4F33@ berechnen. Für ϕ:n(n,n) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dOMaaiOoaiaad6gacqWIMgsycaGGOaGaamOBaiaacYcacaWGUbGaaiykaaaa@3E8E@ etwa erhält man daher für alle n, m (mit k=max{n,m} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iGac2gacaGGHbGaaiiEaiaacUhacaWGUbGaaiilaiaad2gacaGG9baaaa@3EC8@ ):

i=0 n j=0 m a ij i=0 k j=0 k a ij i=0 j=0 a ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6A7E@

Mit diesem Ergebnis gelingt es nun, [5.9.13] auf Doppelreihen zu übertragen.

Bemerkung:  

Jede absolut konvergente Doppelreihe ist konvergent. [5.10.25]

Beweis:  Da | a ij |± a ij 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaakiaacYhacqGHXcqScaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaGccqGHLjYScaaIWaaaaa@44DF@ , sind ( i=0 n j=0 m 1 2 (| a ij |+ a ij ) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcacaGG8bGaamyyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaiabgUcaRiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIWaaabaGaamyBaaqdcqGHris5aaWcbaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@515C@ und ( i=0 n j=0 m 1 2 (| a ij | a ij ) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcacaGG8bGaamyyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaiabgkHiTiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIWaaabaGaamyBaaqdcqGHris5aaWcbaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@5167@ Doppelreihen mit positiven Gliedern. Die nach [5.10.24] gültigen Abschätzungen

i=0 n j=0 m 1 2 (| a ij |± a ij ) i=0 n j=0 m 1 2 (| a ij |+| a ij |) = i=0 n j=0 m | a ij | i=0 j=0 | a ij | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@97B4@

garantieren (wieder mit [5.10.24]) deren Konvergenz, und damit gemäß [5.10.14] auch die Konvergenz von

( i=0 n j=0 m a ij )=( i=0 n j=0 m 1 2 (| a ij |+ a ij ) )( i=0 n j=0 m 1 2 (| a ij | a ij ) ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@80E5@ .

 

Bemerkung:  Ist die Reihe ( i=0 n j=0 m | a ij | ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaGG8bGaamyyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaaWcbaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4916@ beschränkt, so gilt:

i=0 j=0 a ij = j=0 i=0 a ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaadaaeWbqaaiaadggadaWgaaWcbaGaamyAaiaaykW7caWGQbaabeaaaeaacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaWcbaGaamyAaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakiabg2da9maaqahabaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamyAaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoaaSqaaiaadQgacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdaaaa@58DD@
[5.10.26]

Beweis:  Mit ( i=0 n j=0 m | a ij | ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaGG8bGaamyyamaaBaaaleaacaWGPbGaaGPaVlaadQgaaeqaaOGaaiiFaaWcbaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4916@ sind auch ihre Zeilen- und Spaltenreihen monoton wachsend und beschränkt, also konvergent ([5.10.18] bzw. [5.7.1]). Nach [5.10.25] und [5.9.13] trifft dies auch auf die Reihe ( i=0 n j=0 m a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaqahabaWaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaaMc8UaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@4701@ zu. Die Behauptung folgt daher aus [5.10.23].


5.9. 5.11.