Well, if the sequence does have a limit, , say, then, by continuity we should have that
But the equation has only one solution, namely .
Of course, this does not prove that the sequence, starting with , actually does converge to , but it shows that if the sequence converges at all, it converges to .
That the sequence really does converge follows, for example, from Banch's Fixed Point Theorem (aka. Contraction Mapping Theorem), since is contracting.