Prove this sequence converges

**Em Yeu Anh** · February 6th 2010, 11:16 AM

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.

**tonio** · February 6th 2010, 12:12 PM

Originally Posted by Em Yeu Anh

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

**felper** · February 6th 2010, 12:18 PM

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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