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.

You can also use Chebyshev's inequality because if $x\geq y$ then $\frac{1}{x} \leq \frac{1}{y}$. So, we have: $$(x+y)(\frac{1}{x}+\frac{1}{y})\geq2(x\cdot\frac{1}{x}+y\cdot\frac{1}{y})=4$$ 
