Question 3
Suppose $x_1,x_2,\ldots,x_n$ are real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that for any integer $k\gt 1$ there are integers $e_i$ not all $0$ and with $|e_i|\lt k$ such that
$$|e_1x_1+e_2x_2+\cdots+e_nx_n|\le\frac{(k-1)\sqrt n}{k^n-1}.$$
$$|e_1x_1+e_2x_2+\cdots+e_nx_n|\le\frac{(k-1)\sqrt n}{k^n-1}.$$