2010 Putnam A6

Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function

such that $\lim_{x\to\infty} f(x) = 0$. Prove that

$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

令 $f:[0,\infty)\to \mathbb{R}$ 为严格递减连续函数

使得 $\lim_{x\to\infty} f(x) = 0$。证明

$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ 发散。