CGMO 2018 Problem 1

Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$ , for all $n\ge 1$ , where $e$ is the natural logarithm. Prove that for any natural $n$ , $x_n\ge 0$ .