Question 116
Let $n$ and $k$ be positive integers. Prove that for $a_{1}, \ldots, a_{n} \in [1,2^{k}]$ one has
$$
\sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n}
$$
$$
\sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n}
$$