<s>[INST] How to demonstrate this inequality? I have this statement:

If $x > 0 , y > 0$, prove that $\frac{1}{x}+\frac{1}{y} > \frac{2}{x+y}$

I will get the hypothesis? (That's what my teacher calls him, I do not know if it's correct)
$\frac{1}{x}+\frac{1}{y} - \frac{2}{x+y}> 0$
$\frac{x^2 + y^2 + xy + xy - 2xy}{(x+y)xy} > 0$
The hypothesis: $\frac{x^2 + y^2}{(x+y)xy} > 0$
Now I am going to prove, i mean, I need to go back to the origin, which is:
$ \frac {1} {x} + \frac {1} {y}> \frac {2} {x + y} $
Here is my problem, and it is hard for me to go back to the origin, step by step, could you help me? I have 5 sheets worn.
PD: I tried this:

 [/INST] Use a better hypothesis, and get a better inequality.
\begin{align}
\frac{(x-y)^2}{(x+y)xy} &\ge 0 \\
\frac{x^2 - 2xy + y^2}{(x+y)xy} &\ge 0 \\
\frac{x^2 + 2xy + y^2}{(x+y)xy} &\ge \frac{4xy}{(x+y)xy} \\
\frac{(x+y)^2}{(x+y)xy} &\ge \frac{4}{x+y} \\
\frac{x+y}{xy} &\ge \frac{4}{x+y} \\
\frac{1}{x} + \frac{1}{y} &\ge \frac{4}{x+y} > \frac{2}{x+y} \\
\end{align}
</s>