Limit of a recursively defined sequence

I'm having some trouble with this question.

Suppose that and whenever

1) Show that whenever .

2) Use induction to show that the sequence is bounded.

Would I do the question like this:

Prove that

Assume , then

By induction it follows that is bounded above by 2.

3) Use induction to show that is an increasing sequence

Prove

Assume that

By induction it follows that, is an increasing sequence

4) Explain why exists

Since is a monotonically increasing sequence, bounded above, it must converge to a limit as

5) Find . (That is, find

Can someone check whether what I did is correct and help me with (1) and (5)