CJMO 2020 Problem 1

Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$ . Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$ .