<s>[INST] Prove that $\frac{1}{1 - \sqrt{ab}} + \frac{1}{1 - \sqrt{bc}} + \frac{1}{1 - \sqrt{ca}} \leq \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ Given that $0 < a , b , c < 1$.
Prove that $\frac{1}{1 -  \sqrt{ab}} + \frac{1}{1 - \sqrt{bc}} + \frac{1}{1 - \sqrt{ca}} \leq \frac{1}{1 -  a} + \frac{1}{1 - b} + \frac{1}{1 - c}$.
I tried using modified C.S. and brute-force. But , it demands a lot of calculation. So , I want some better solution than this. Thank you.
 [/INST] The RHS is
$$\sum_{n=0}^\infty(a^n+b^n+c^n).$$
The LHS is
$$\sum_{n=0}^\infty(\sqrt{a^nb^n}+\sqrt{a^nc^n}+\sqrt{b^nc^n}).$$
For each $n$,
$$a^n+b^n+c^n=\frac{a^n+b^n}2+\frac{a^n+c^n}2+\frac{b^n+c^n}2
\ge\sqrt{a^nb^n}+\sqrt{a^nc^n}+\sqrt{b^nc^n}$$
by AM/GM
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