Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$

Question

Help I'm really stuck on a proof question:

Prove that, for all positive integers for $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$

Thanks!

Answer 1

Start with $(\sqrt{x} - \sqrt{y})^2 \geq 0$.

Answer 2

\begin{eqnarray*} (\sqrt{x}- \sqrt{y})^2 \geq 0 \end{eqnarray*} and now do a little bit of algebra.