BALTIC-WAY 2015 Problem 2

Let $n$ be a positive integer and let $a_{1}, \ldots, a_{n}$ be real numbers satisfying $0 \leq a_{i} \leq 1$ for $i=1, \ldots, n$. Prove the inequality

$$\left(1-a_{1}^{n}\right)\left(1-a_{2}^{n}\right) \cdots\left(1-a_{n}^{n}\right) \leq\left(1-a_{1} a_{2} \cdots a_{n}\right)^{n} .$$