Question 18
Let $x_{1}, x_{2}, \ldots, x_{n}$, real numbers that meet $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$. Demonstrate that for each integer $k \geq 2$ there are integers not all null $a_{1}, a_{2}, \ldots, a_{n}$, such that $\left|a_{i}\right| \leq k-1$ for all $i$ and $$
\left|a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\right| \leq \frac{(k-1) \sqrt{n}}{k^{n}-1}
$$
\left|a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\right| \leq \frac{(k-1) \sqrt{n}}{k^{n}-1}
$$