I am having difficulty proving that \begin{align} \sum_{i = 0}^{M-1} \sum_{j = 0}^{N - 1} \delta \left[a - \left(Ni + j\right)\right] = 1 \end{align} for $M < N$ and $0 \leq a < NM$. Where $\delta[n] = 1$ for $n = 0$ and $\delta[n] = 0$ otherwise. I guess I am mainly struggling with how to concisely show that $Ni + j$ takes on every value from $[0,1,\ldots,NM-1]$ exactly once during the double summation.
How does one show this convincingly starting with the original summation? I was reading a paper on non-uniform sampling theory and a similar expression with Dirac delta distributions where used, but the argument provided was mostly qualitative and I was looking for something with a little more detail.