1994 Putnam A1

Suppose that a sequence $a_1, a_2, a_3, \dots$ satisfies

$0 < a_n \leq a_{2n} + a_{2n+1}$ for all $n \geq 1$. Prove that the series

$\sum_{n=1}^{\infty} a_n$ diverges.

假设序列 $a_1, a_2, a_3, \dots$ 满足

$0 < a_n \leq a_{2n} + a_{2n+1}$ 对于所有 $n \geq 1$。证明该系列

$\sum_{n=1}^{\infty} a_n$ 发散。