I am currently working through some notes and I'm a little confused on how the following equality is derived: $$\sum_{i=0}^\infty \sum_{j=0}^\infty \left( \frac{1}{(k+i+j+1)(k+i+j+m)} - \frac{1}{(k+i+j+1)(k+i+j+m+1)}\right) = \sum_{l=1}^\infty \frac{l}{k+l}\left( \frac{1}{(k+l+m-1)} - \frac{1}{(k+l+m)}\right)$$
It looks as if the following substitution has been made: $$ i+j+1 =l $$ However I'm a little confused where the l in the numerator comes from in the final line.
It comes from counting up how many times $i+j+1$ can take the value $l$. Since $i,j\geq0$, $i$ can take the values $0,1,\dots,l-1.$