# Prove $\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}}$

For postive integer $a, b, c$ prove the following inequality $$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} > \sqrt{2}$$ How we can prove this inequality?

• The title is missing a piece and the question body too: what are the assumptions on $a,b,c$? – Jack D'Aurizio Sep 21 '15 at 13:48
• Please provide the missing parts. – Kushal Bhuyan Sep 21 '15 at 13:53
• @AdityaAgarwal Not dealing with limits here. $1/0$ is undefined. – Gummy bears Sep 21 '15 at 14:00
• That is what I am saying, there have to be some assumptions. – Aditya Agarwal Sep 21 '15 at 14:00
• @AdityaAgarwal He's obviously missin the sum of a, b, c – Gummy bears Sep 21 '15 at 14:01

$$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} > \sqrt{2}$$ By$AM-GM$ $$\sum_{cyc}\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}}=\sum_{cyc}\frac{\sqrt2}{2\cdot\frac{1}{\sqrt2}\sqrt{b+\frac{1}{a}+\frac{1}{2}}}\geq\sum_{cyc}\frac{\sqrt2}{b+\frac{1}{a}+1}=\sqrt2$$