Prove that a recursive sequence is monotonically increasing $a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$
Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$
If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it.

For the first part, $a_1, a_2$ are less than four. You want to use something usually called "strong induction." That is, suppose that for all $k \le n$ we have $a_k \le 4$. Then $$a_n \le \frac{4}{2} + \sqrt{4} = 4.$$
For the second again use strong induction, note that 
$a_{n+1} = \frac{1}{2}a_n + \sqrt{a_{n-1}}$ 
and by induction $a_n \ge a_{n-1}$ and $a_{n-1} \ge a_{n-2}$ which gives us
$$a_{n+1} \ge \frac{1}{2} a_{n-1} + \sqrt{a_{n-2}} = a_n.$$
