Prove convergence of $x_{n+1} = \frac{x_n}{2} + \frac{2}{x_n}$ Given $x_1= 1$ and  $$x_{n+1} = \frac{x_n}{2} + \frac{2}{x_n}$$ for all $n\in\mathbb{N}, n\geq 1$.  Prove that the sequence $\{x_n\}$ converges and find its limit.
The hint given says that taking the limit of both sides is only valid if you know the limit exists already. 
Part a of the question determines that $ab \leq \frac{a^2 + b^2}{2}$.
I assume we try to use Cauchy along with a trick from part (a)? However, I have no clue how to go about that.  Thank you!

as pointed out by Doug M, $x_1 = 5/2$ which is above $2$
Once $x > 2,$ we have $$ 0 < \frac{x-2}{2x} < \frac{1}{2} $$
The interesting thing is how $x-2$ shrinks under your mapping:
$$ \frac{x}{2} + \frac{2}{x} - 2 = \left(\frac{x-2}{2x} \right) (x-2) , $$
so with $x > 2$ we get
$$ \frac{x}{2} + \frac{2}{x} - 2 < \left(\frac{1}{2} \right) (x-2) , $$
with $n \geq 1$ we have
$$ (x_{n+1} - 2 ) < \frac{1}{2} (x_n - 2)  $$
You don't need the next thing: there is actually quadratic convergence.
