Proving convergence of a series and then finding limit I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$
converges and find the limit. 
I started by defining the sequence by $x_1=\sqrt{2}$ and then $x_{n+1}=\sqrt{2+x_n}$. Then I proved by induction that the sequence is increasing and that it is bounded. Then I use the Monotone Convergence Theorem to prove that it converges.
Now, I claim that the limit of the sequence is $2$. So I need to show that for all $\epsilon>0$, there exists $N$ such that for all $n\geq N$ we have $|x_n-2| < \epsilon$. This is kind of where I'm stuck. I don't know how to proceed to prove the convergence to $2$. 

Prove that this sequence is increases and bounded. Hence it convergence to some point $p\in \mathbb{R}$ and make limit transition in $x_{n+1}=\sqrt{2+x_n}$
