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Math Help - Prove this sequence converges

  1. #1
    Member Em Yeu Anh's Avatar
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    Question Prove this sequence converges

    Suppose that a_1 = 1 and a_n = \sqrt{2+a_{n-1}} for all  n \geq 2. Prove that this sequence converges and find its limit.

    I was hoping that I could avoid induction on this one but I don't know any other method to prove this.
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  2. #2
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    Quote Originally Posted by Em Yeu Anh View Post
    Suppose that a_1 = 1 and a_n = \sqrt{2+a_{n-1}} for all  n \geq 2. Prove that this sequence converges and find its limit.

    I was hoping that I could avoid induction on this one but I don't know any other method to prove this.

    Yes , induction is the key here. Prove that this is monotone increasing sequence and that a_n\leq 2\,\,\,\forall n\in\mathbb{N}

    Tonio
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  3. #3
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    You can prove that the sequence is crecent and is bounded. To calculate the limit, you can use that the subsequences of a sequences converges to the same limit. Then, L=\sqrt{2+L}
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