Is it possible to swap sums like that? Say that I have two sums like this :
$$\sum_{a=0}^n\sum_{b=0}^m f_{ab}$$
Would it be true to say that this expression can be considered as equal to :
$$\sum_{a=0}^m\sum_{b=0}^n f_{ab}$$
As long as the expression that comes after the sums is the same is both cases ? If it is true, is it easy to prove ?  
Thank you !
EDIT :
This is what I want to prove :


Not at all, look:
$$\sum_{a=1}^2\sum_{b=1}^3 \frac ab=\sum_{a=1}^2 \left(\frac a1+\frac a2+\frac a3\right)=\frac 11+\frac 12+\frac 13+\frac 21+\frac 22+\frac 23=\frac {11}2,$$
and
$$\sum_{a=1}^3\sum_{b=1}^2 \frac ab=\sum_{a=1}^3 \left(\frac a1+\frac a2\right)=\frac 11+\frac 12+\frac 21+\frac 22+\frac 31+\frac 32=9.$$
You can swap the sums, but not just what is on the top of your sum.
You do have for all finite sums:
$$\sum_{a=0}^n\sum_{b=0}^m f_{ab}=\sum_{b=0}^m\sum_{a=0}^n f_{ab}.$$
Edit
What you want to prove works because you are dealing with finite sums , and you changed the indices too.
