MathDB

Problem 3

Part of 2019 CIIM

Problems(1)

CIIM 2019 Problem 3

Source:

8/30/2021
Let {an}n∈N\{a_n\}_{n\in \mathbb{N}} a sequence of non zero real numbers. For m≥1m \geq 1, we define: Xm={X⊆{0,1,…,m−1}:∣∑x∈Xax∣>1m}. 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→∞∣Xn∣2n=1.\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.
CIIM