Question 29
Let $(x_n)_{n \ge 1}$ be an increasing unbounded sequence of natural numbers such that $x_1 = 1$ and $x_{n+1} \le 2x_n$ for all $n \ge 1$. Prove that every nonzero natural number can be written as a finite sum of pairwise distinct terms of the sequence $(x_n)_{n \ge 1}$.
*Note:* Two terms $x_i$ and $x_j$ of the sequence $(x_n)_{n \ge 1}$ are said to be distinct if $i \neq j$.
*Note:* Two terms $x_i$ and $x_j$ of the sequence $(x_n)_{n \ge 1}$ are said to be distinct if $i \neq j$.