Evaluation of an indexed sum.

Question

I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds.

$\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$

This is very simple I know, but for some reason I'm unclear on this.

Answer 1

Let's unpick the sum.

On the outside, we have $\sum_{m = 0}^0$ which means this sum has only one term, which is $m = 0$.

Then inside that, we have $\sum_{k = 0}^m$, but since we know that $m = 0$ is the only possible value, this reduces to $\sum_{k = 0}^0$, which again is a single term of $k = 0$.

So now the complete sum is $\sum_{m = 0}^0 \sum_{k = 0}^m a_{m-k, k} = a_{0-0, 0} = a_{0,0}$.