Assistance on a recursive proof

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The discussion revolves around proving that the sequence defined by A0 = 2 and An+1 = An/2 + 1/An satisfies the inequality An ≤ √(2) + 1/2n for all n ≥ 0. Participants agree that mathematical induction is the appropriate method for the proof. There is a request for assistance with the inductive step, indicating some difficulty in formulating it. The conversation highlights the importance of applying the recursive definition directly into the proof structure. Overall, the focus remains on clarifying the inductive approach to validate the sequence's properties.
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Homework Statement



Define a sequence of numbers Ai by: A0 = 2, An+1 = An/2 + 1/An (for n greater than or equal to 1). Prove that An less than or equal to √(2) + 1/2n for all n greater than or equal to 0. I think it's a safe bet that induction should be used here. I'm having trouble finding the inductive step though maybe just cause I'm not feeling particularly well today. Help is always appreciated.

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The Attempt at a Solution

 
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hi black_hole! :smile:
black_hole said:
I think it's a safe bet that induction should be used here.

yes …

just put it into the equation! :rolleyes:​
 

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