A Series Convergence Problem - Cont'd

This is a continuation of the last post.

4. Looking back
I went and slept on the problem, and I'm not exaggerating, woke up with an epiphany about the problem. All night I dreamt about series and floating sigma signs. Other than the sheer weirdness of waking up with a solution to a math problem, especially one that wasn’t that hard, I did realize that there was a much simpler and more intuitive solution to the problem! The approach that I wrote up in the last post is perhaps a little brute-forcing the way to a solution. This new solution uses an evident truth about series to prove the condition in one step. Here:
  Assuming a_n converges.
            Since (SIGMA)a_n converges, then ((SIGMA)a_n)^2 must also converge.
            Since a_n => 0, then ((SIGMA)a_n)^2 >= (SIGMA)a_n^2 >= 0.
            Then by the pinching theorem, (SIGMA) a_n^2 converges.
Then, given that a_n converges, then (a_n)^2 must converge.

Solutions are either valid or invalid, but I somehow feel this is a better solution. If only you had the time to sleep on every problem in the world…