The possible finite limits of the sequence are the solutions of the equation...
... that are...
A quickly inspection reveals that if is...
Yes, that is limit. But how to prove convergence of this, using therem of monoton and bounded sequences? It's not quite easy, it must be considered, I think, subsequences
One is increasing, second decreasing, and both oscilate about . This may be a strict proof.
It is not at all difficult to prove that...
On the basis of these unequalities we have the following distinct cases...
a) if the sequence diverges at
b) if is ,
c) if the sequence converges at in any case with 'oscillatory behavior' unless is that produces the constant sequence , ...
and you can "see " that , and "is" monotonicaly increasing, and , and "is" monotonicaly decreasing.
So, here is neccessary only strict proof of this monotonity. Maybe this relation
Observe that , (by inicial value) for all , so second parenteze is negative, and somehow here must be used induction.
I belive similar can be done for
Update: Hear is another relation
, so this could be used maybe (I think) for variant of induction (with assumptation for and , and proff for and ).