Question 45
Suppose positive number sequence $\{x_n\}$ satisfies $S_n \ge 2S_{n-1}$, $n = 2, 3, \dots$, where $S_n = x_1 + \dots + x_n$. Prove that there exists a constant $C \gt 0$, such that
$$
x_n \ge C \cdot 2^n,\ n = 1, 2, \dots
$$
$$
x_n \ge C \cdot 2^n,\ n = 1, 2, \dots
$$