Proving Proving $ \sqrt{a \cdot b} \le (a+b)/2$ with a and b in N*

Question

We have variables $a$ and $b$ as natural numbers..

I tried using the reccurence but I got stuck proving:

$\sqrt{(a+1) \cdot (b+1)} \le (a+1+b+1)/2$

May someone help me with this?

Answer 1

$ (\sqrt{a}-\sqrt{b})^2 \ge 0$ and $(\sqrt{a}-\sqrt{b})^2=a-2 \sqrt{ab}+b.$

Can you proceed ?