$\sum a_n$ converges $\implies\ \sum a_n^2$ converges? 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. 

If $\sum_{n=1}^\infty a_n$ is convergent, then $\lim_{n\to\infty}a_n=0$, hence from somewhere upward we have $0<a_n<1$, now use comparison test...
