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## Homework Statement

Let ##f: (1, \infty) \to [0,\infty)## be a function such that the improper integral ##\int_{1}^{\infty} f(x)dx## converges. If ##f## is monotonically decreasing, then ##\lim_{x \to \infty} f(x)## exists.

## Homework Equations

## The Attempt at a Solution

This problem doesn't come from a textbook, so there are no restrictions on what theorems can be used. However, I would prefer an elementary solution. My strategy is to show that if ##f## is not bounded below, then ##\int_{1}^{\infty} f(x)dx## couldn't possibly converge. This would mean that ##f## would have to bounded below in addition to being monotonically decreasing, which I believe would imply that ##\lim_{x \to \infty} f(x)## exists. I could use some hints.