Proof: First we show by induction
which in fact means that is already bounded.
► From we can estimate
and thus have .
Furthermore proves to be decreasing: Because implies , the assertion holds due to the following equivalence:
All in all has a limit, say . But g is the limit of as well, so we can calculate g using the limit theorems:
As g is unique, g satisfies the equation . So we have (note that ):