If $\sum\limits_{n=1}^∞u_n^2$ is convergent, then $\sum\limits_{n=1}^∞\frac{u_n}n$ is absolutely convergent

Question

If $\{u_n\}$ is a sequence of real numbers and the series $\displaystyle\sum_{n=1}^{\infty}u_n^2$ is convergent, prove that the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is absolutely convergent.

I have proved the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is convergent, but how to prove it is absolutely convergent?

Answer 1

One may use$$ 2\cdot \frac{|u_n|}n \le |u_n|^2+\frac{1}{n^2},\quad n\ge1. $$

Answer 2

Apply Cauchy-Schwarz:$$\sum_{i=1}^N\left|\frac{u_n}n\right|\leqslant\sqrt{\sum_{i=1}^n{u_i}^2}\sqrt{\sum_{i=1}^n\frac1{i^2}}.$$