ISL 2019 A2

Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying $$u_{1}+u_{2}+\cdots+u_{2019}=0 \text { and } u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 .$$ Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that $$a b \leqslant-\frac{1}{2019}$$ (Germany)