1. 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.

2. 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

3. 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, $L=\sqrt{2+L}$