I am trying to prove that if $a_n$ >= 0 and $\sum a_n$ converges. Show that $\sum a_n^2$ converges. I'm attempting to use the comparison test by setting it up like, 0 <= $\sum a_n$ <= $\sum a_n^2$. I can do this by showing that $ a_n^2$ goes to zero faster than $a_n$, thus all the points in $a_n^2$ <= $a_n$. If that is true then $\sum a_n^2$ <= $\sum a_n$. Thus, by comparison test the $\sum a_n^2$ is convergent.
I feel like there is a counter example to my argument. Although I haven't been able to come up with one. Is there an example where a sequence converges to zero faster than it's square?