How can I Prove $\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$ for $x,y>0$ prove that
$\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$
I have tried to develop $(x+y)^2=$ and to get to an expression that must be bigger than those above
Thanks!

$\sqrt{xy} \le \dfrac{x + y}{2}$ is just the AM–GM inequality. $\dfrac{2xy}{x + y} \le \sqrt{xy}$ is also the AM-GM inequality applied to $1/x$, $1/y$.
