If $x,y$ are positive, then $\frac1x+\frac1y\ge \frac4{x+y}$ For $x$, $y$ $\in R^+$, prove that $$\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}$$
Could someone please help me with this inequality problem?  I have tried to use the AM-GM inequality but I must be doing something wrong. I think it can be solved with the AM-GM but I can’t  solve it. Thanks in advance for your help.

Here is a solution with AM-GM:
$$\frac{1}{x}+\frac{1}{y} \geq \frac{2}{\sqrt{xy}}$$
$$x+y \geq 2 \sqrt{xy} \Rightarrow \frac{1}{\sqrt{xy}} \geq \frac{2}{x+y}\Rightarrow \frac{2}{\sqrt{xy}} \geq \frac{4}{x+y}$$
Also you can note that
$$(x+y)(\frac{1}{x}+\frac{1}{y}) \geq 4$$
is just Cauchy-Schwarz.
