TITLE: If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $
QUESTION [16 upvotes]: Problem: 

Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that
  $$\sum_0^\infty a_n$$
  converges, show that 
  $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$
  converges.

Source: Rudin, Principles of Mathematical Analysis, Chapter 3, Exercise 7.

REPLY [22 votes]: We have for all real numbers $2ab\leq a^2+b^2$ hence 
$$0\leq \frac{\sqrt{|a_n|}}n\leq \frac{|a_n|+\frac 1{n^2}}2.$$
Since the series $\sum_n|a_n|$ and $\sum_n\frac 1{n^2}$ are convergent, we get the convergence of $\sum_n\frac{\sqrt{|a_n|}}n$.