e  is irrational


Suppose e to be rational. Then there is a representation e= n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzaiabg2da9maalaaabaGaamOBaaqaaiaad2gaaaaaaa@394E@ with n,m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGHxiIkaaaaaa@3C0A@ . For the integer

(e i=0 m 1 i! )m!= nm! m i=0 m m! i! =n(m1)! i=0 m (i+1)(i+2)m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@748A@        [+]

we calculate the following estimate using the limit of the geometric series:

0<(e i=0 m 1 i! )m! = i=0 m! i! i=0 m m! i! = i=m+1 m! i! = i=m+1 1 (m+1)(m+2)i i=m+1 1 (m+1) im = i=0 1 (m+1) i+1 = 1 m+1 1 1 1 m+1 = 1 m <1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B1C2@

According to [+] this proves the existance of an integer within ]0,1[ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiyxaiaaicdacaGGSaGaaGymaiaacUfaaaa@394E@ .   Contradiction!