Stability of the summation function
Well, we are summing over $k$ in the range $n_0$ to $n$. The number of elements in the sum is therefore $n-n_0$. We also know that the sum must be positive and that all terms in the sum ($x[n]$) have absolute value less than $M$.

To account for $n-n_0$ being negative, but our sum MUST be positive, we take its absolute value.