This is problem 903 from The College Mathematics Journal, v. 41, n 3, May 2010. The problem was proposed by José Luis Díaz-Barrero and José Gibergans-Báguena, Barcelona, Spain.

Prove that

\Bigg(\sum^{n}_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}},

where n is a positive integer.

By the Cauchy-Schwarz inequality and telescoping

\Bigg(\sum^{n}_{k=1}1\cdot\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le \Bigg(\sum^{n}_{k=1}1^{2}\Bigg)\Bigg(\sum^{n}_{k=1}\frac{k-\sqrt{k-1}\sqrt{k+1}}{\sqrt{k}\sqrt{k+1}}\Bigg)
= n\sum^{n}_{k=1}\Bigg(\sqrt{\frac{k}{k+1}}-\sqrt{\frac{k-1}{k}}\Bigg)=n\sqrt{\frac{n}{n+1}}\cdot