lower bound on $xy/(x+y)?$ I am looking forward to a lower bound on $x*y/(x+y),$ where both of $x\geq 1$ and $y\geq 1$ are real.
The upper bound on $x*y/(x+y)$ is obvious, for example $x$ or $y$. But a lower bound is not very obvious to me. I want a simple lower bound, which is not very complex.
Could anyone pass some wisdom to me? 
Thanks a lot in advance..

$$\frac{xy}{x+y}=\frac{1}{\frac{1}{x}+\frac{1}{y}}$$
So to compute lower bound on this expression is same as computing upper bound on $\frac{1}{x}+\frac{1}{y}$
Since, $x\geq 1\implies \frac{1}{x}\leq1$ and similarly $\frac{1}{y}\leq 1$
Thus, $$\frac{1}{x}+\frac{1}{y}\leq 2 $$ $$\implies \frac{xy}{x+y}\geq \frac{1}{2}$$
