CIIM 2019 Problem 3

Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers. For $m \geq 1$ , we define: $$X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}.$$ Show that $$\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.$$