Let $A\ne\{0\}$ be a ring with unity and $p,q,r\in A^{\mathbf{N}}$. I want to show that $$\sum_{j=0}^n\sum_{k=0}^jp_kq_{j-k}r_{n-j}=\sum_{k=0}^n\sum_{j=k}^np_kq_{j-k}r_{n-j}$$ for all $n\in\mathbf{N}$.
Attempt:
I am trying to prove this by induction. For $n=0$, it holds trivially. Let it hold for some $n\in\mathbf{N}$. I am having trouble with the induction step. We have
$$\begin{align} \sum_{k=0}^{n+1}\sum_{j=k}^{n+1}p_kq_{j-k}r_{(n+1)-j}&=\sum_{k=0}^n\sum_{j=k}^{n+1}p_kq_{j-k}r_{(n+1)-j}+\sum_{j=n+1}^{n+1}p_{n+1}q_{j-(n+1)}r_{n+1-j}\\ &=\sum_{k=0}^n\sum_{j=k}^{n}p_kq_{j-k}r_{(n+1)-j}+p_{n+1}q_{j-(n+1)}r_0+p_{n+1}q_{0}r_{0} \end{align}.$$ It seems to me that the problem is the expression $r_{(n+1)-j}$ since it's preventing me from applying the induction hypothesis to the expression $\sum_{k=0}^n\sum_{j=k}^{n}p_kq_{j-k}r_{(n+1)-j}$. But I'm not sure.
Any hints?
It's not preventing you. Let $s_h=r_{h+1}$. Then your inductive hypothesis tells you that $$ \sum_{j=0}^n\sum_{k=0}^jp_kq_{j-k}r_{n+1-j}=\sum_{j=0}^n\sum_{k=0}^jp_kq_{j-k}s_{n-j}=\sum_{k=0}^n\sum_{j=k}^np_kq_{j-k}s_{n-j}=\sum_{k=0}^n\sum_{j=k}^np_kq_{j-k}r_{n+1-j}. $$ On a separate note, what you need to prove is simply the equality $$ \{(k,j):\ 0\leq j\leq n,\ 0\leq k\leq j\}=\{(k,j):\ 0\leq k\leq n,\ k\leq j\leq n\}, $$ as all you are doing is exchanging the sums. This has nothing to do with convolution.