Question 56
Let $a_{1}, \ldots, a_{n}$ be a non increasing sequence of positive real numbers. Prove that
$$
\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} \leq a_{1}+\frac{a_{2}}{\sqrt{2}+1}+\cdots+\frac{a_{n}}{\sqrt{n}+\sqrt{n-1}}
$$
When does equality hold?
$$
\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}} \leq a_{1}+\frac{a_{2}}{\sqrt{2}+1}+\cdots+\frac{a_{n}}{\sqrt{n}+\sqrt{n-1}}
$$
When does equality hold?