Question 65
Let $a_1, \dots, a_n$ be real numbers, fulfilling $0 \le a_i \le 1$ for $i = 1, \dots, n$. Prove the inequality
$$
(1 - a_1^n)(1 - a_2^n) \cdots (1 - a_n^n) \le (1 - a_1 a_2 \cdots a_n)^n.
$$
$$
(1 - a_1^n)(1 - a_2^n) \cdots (1 - a_n^n) \le (1 - a_1 a_2 \cdots a_n)^n.
$$