- #1

- 1,462

- 44

## Homework Statement

Prove that if ##(a_n)## is a decreasing sequence of positive numbers and ##\sum a_n## converges, then ##\lim na_n = 0##

## Homework Equations

## The Attempt at a Solution

Let ##\epsilon >0##. By the Cauchy criterion there exists an ##N\in \mathbb{N}## such that ##\forall n\ge m\ge N##, we have that ##|\sum_{k=m+1}^{n}a_k|<\epsilon##. But the sequence is decreasing, so ##|(n-m)a_n|\le |\sum_{k=m+1}^{n}a_k|<\epsilon##. So we have that ##|na_n-ma_n|<\epsilon## for all ##\epsilon>0##. So ##na_n=ma_n## for all ##n\ge m\ge N##. Since the tails of these sequence are the same eventually, they have the same limit. Since ##\sum a_n## converges, we see that ##\lim ma_n = 0##. Hence ##\lim n a_n = 0##.