<s>[INST] Prove that $\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$ $$\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$$
I tried applying a.m. g.m inequality to l.h.s and tried to find upper bound for l.h.s and lower bound for r.h.s but i am not getting answer . 
 [/INST] Not a beautiful proof I have to admit, but it is a proof:
Multiply 2 sides of equation with $(a+b)(c+d)(a+b+c+d)$, we get to the equivalent form:
$$
ab(c+d)(a+b+c+d) + cd(a+b)(a+b+c+d) \leq (a+c)(b+d)(a+b)(c+d).
$$
Work out both sides of the equation above to reduce it to the following equivalent form:
$$
2abcd \leq a^2d^2 + b^2c^2,
$$
which is obviously true due to the AM-GM inequality.
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