Show that if $\sum_n |a_n|$ converges, then $\sum_n a_n^2$ converges 
Show that if $\sum_n |a_n|$ converges, then $\sum_n a_n^2$ converges
  whenever all $a_n$ are in $\mathbb{R}$.

Lemma: I have already proved that if $\sum_n a_n$ and $\sum_n b_n$ converge absolutely, then $\sum_{n,m} a_n b_m$ converges absolutely.
Let's take $b_n:=a_n$ for all $n$. From the above lemma we know that $\sum_{n,m} a_n a_m$ converges absolutely. Since $\sum_n a_n^2 \le \sum_{n,m} |a_n a_m|$, by comparison we get that $\sum_n a_n^2$ is convergent.
Now I have two questions:
1) Is the above derivation correct?
2) Is it possible to deduce this without refering to the Lemma?

Addressing only question 2: If $\sum|a_n|$ converges, then $|a_n|\to0$ so eventually we must have $|a_n|<1$. Let us assume (Wlog) that $|a_n|<1$ for all $n$. Then $a_n^2=|a_n|^2<|a_n|$ for all $n$ so we can apply the comparison test.
