Hey, I'm stuck on this question and desperately need help.

Question

Suppose that the sequence is convergent, and assume that the sequence is bounded. Prove that the sequence defined by

is convergent and find its limit. (n belongs to the set of natural numbers)

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Please help! I tried using the theorem that if a sequence is convergent, then it is bounded (for ) but get anywhere. It's quite clear to me that the limit is but I don't know how to get there.