Inductively define a sequence by and . Show that converges.

I'm having trouble proving this, my idea was to show that this sequence was Cauchy and hence convergent(since we're in ). However I'm having difficulty showing its Cauchy as for arbitrary , I need where this holds for some integer , and . My problem is with how to choose an such that holds for arbitrary .

Also another similar question:

Inductively define a sequence by and . Prove converges and evaluate the limit.

My idea to show that it converges was to show that its monotonically decreasing and bounded. And then I know the limit is 2, since I played around with it on my calculator. However I had problems when I tried to use induction to show that its monotonically decreasing, the equations get really messy and I never get anything nice I can use.

Also a similar things happens when, for abitrary I try find such that , . The definition of is not in terms of , so I'm not sure how to find an that meets the requirement.

Any hints would be greatly appreciated.