Double summation with infinite limit, change of index using Iverson bracket notation I'm trying to understand the change of variable for this double summation:
$$\sum_{n=0}^{\infty} \sum_{m=0}^{n} \xi(m,n)=\sum_{\ell=0}^{\infty} \sum_{m=0}^{[\frac{\ell}{2}]} \xi(m,n=\ell-m)$$
where $\ell=m+n$ and $[\frac{\ell}{2}]$ means summing up to the largest integer that is less or equal than $\frac{\ell}{2}.$ So based on this question Double summation, index change clarification. I was trying to use Iverson bracket notation but I'm not sure of my deduction.
What I have so far is
$$\begin{align}\sum_{n=0}^{\infty} \sum_{m=0}^{n} \xi(m,n) &= \sum_{n,m} [0\leq n<\infty]\cdot [0\leq m \leq n] \xi(m,n) \\ &= \sum_{n,m} [0\leq m \leq n < \infty] \xi(m,n) \end{align}$$
From here I could write
$$\sum_{n,m} [0\leq m <\infty]\cdot [m\leq n < \infty] \xi(m,n) =  \sum_{m=0}^{\infty} \sum_{n=m}^{\infty} \xi(m,n)$$
So to get my result I'm not sure if I can also write
$$\begin{align}[0\leq m \leq n < \infty] &\rightarrow [0\leq 2m \leq n + m < \infty] \\ &\rightarrow [0\leq 2m \leq \ell < \infty]\cdot [\ell=m+n] \\ &\rightarrow [0\leq m \leq \frac{\ell}{2}]\cdot[0 \leq \ell < \infty]\cdot [\ell=m+n]\end{align}$$
What also confuses me is if in the last line I should write the first bracket as either $2m\leq\ell$ or $2m < \infty$ to separate the conditions.
 A: The work with Iverson brackets looks (besides notational aspects) fine. Note that in the derivation we not only have to show implications $\rightarrow$ but equivalence relations of the index regions we want to sum up.
Here is a derivation based upon index transformations without using Iverson brackets.

We obtain
\begin{align*}
\color{blue}{\sum_{n=0}^{\infty}\sum_{m=0}^n\xi(m,n)}
&=\sum_{0\leq m\leq n<\infty}\xi(m,n)\tag{1}\\
&=\sum_{{0\leq m}\atop{2m\leq m+n<\infty}}\xi(m,n)\tag{2}\\
&=\sum_{{0\leq m}\atop{2m\leq l<\infty}}\xi(m,l-m)\tag{3}\\
&=\sum_{0\leq l<\infty}\sum_{0\leq 2m\leq l}\xi(m,l-m)\tag{4}\\
&\,\,\color{blue}{=\sum_{l=0}^\infty\sum_{m=0}^{\lfloor l/2\rfloor}\xi(m,l-m)}
\end{align*}
and the claim follows.

Comment:

*

*In (1) we write the index region somewhat more conveniently.


*In (2) we do an equivalence transformation of the inequality chain by adding $m$ as preparation for the next step.


*In (3) we introduce $l$ and substitute $l=m+n$.


*In (4) we rewrite the index region, so that summing over $l$ becomes the outer series.
