TITLE: Square of a series
QUESTION [6 upvotes]: Suppose that $\sum_{j=0}^\infty a_j$ converges and $\sum_{j=0}^\infty a_j^2<\infty$, where $a_j\in\mathbb R$, $j\ge0$. I would like to prove that
  $$
\biggl[\sum_{j=0}^\infty a_j\biggr]^2
=\sum_{j=0}^\infty a_j^2+2\sum_{k=1}^\infty\sum_{j=0}^\infty a_ja_{j+k}.
$$

Using continuity and squaring the sum, we obtain
$$
\biggl[\sum_{j=0}^\infty a_j\biggr]^2=\lim_{n\to\infty}\biggl[\sum_{j=0}^na_j\biggr]^2=\sum_{j=0}^\infty a_j^2+2\lim_{n\to\infty}\sum_{j=0}^{n-1}\sum_{k=1}^{n-j}a_ja_{j+k}.
$$
The proof would be complete if I could justify the following two equalities:

$\lim_{n\to\infty}\sum_{j=0}^{n-1}\sum_{k=1}^{n-j}a_ja_{j+k}=\sum_{j=0}^\infty\sum_{k=1}^\infty a_ja_{j+k}$;
$\sum_{j=0}^\infty\sum_{k=1}^\infty a_ja_{j+k}=\sum_{k=1}^\infty\sum_{j=0}^\infty a_ja_{j+k}$.

Are these equalities true? How can I justify them if they're true? Or should the proof be completely different?
Any help is much appreciated!

REPLY [3 votes]: This is justified by showing the limit: $2\lim_{n\to\infty}\sum_{j=0}^{n-1}\sum_{k=1}^{n-j}a_ja_{j+k}$ exists. It exists because the other terms in your equation are finite (i.e. their limits exist).
What if you begin with the sums the desired way around. Then there's no need to swap them at the end. i.e. Instead of:

$\qquad\qquad 2\lim_{n\to\infty}\sum_{j=0}^{n-1}\sum_{k=1}^{n-j}a_ja_{j+k}$
$\qquad$ have:
$\qquad\qquad2\lim_{n\to\infty}\sum_{k=1}^{n}\sum_{j=0}^{n-k}a_ja_{j+k}$