TITLE: $\sum a_n$ converges $\implies\ \sum a_n^2$ converges?
QUESTION [11 upvotes]: If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ always convergent? Either prove it or give a counter example.
Im trying in this way, Suppose $a_n \in [0,1] \ \forall\  n.\ $ Then ${a_n}^2\leq a_n\ \forall\  n.$ Therefore by comparison test $\sum {a_n}^2$ converges.
So If $a_n$ has certain restrictions then the result is true.
 what about the general case?
How to proceed further?
Hints will be greatly appreciated.

REPLY [22 votes]: Hint: Since, $a_i>0\forall i\ge 1$, $$\sum_{n=1}^N a_i^2\le \left(\sum_{i=1}^N a_i\right)^2\ \forall N\ge 1$$