Let $a_{ij} \in [0,\infty)$ for all $(i,j) \in w \times w$, where $w =\{ 0,1,2,...\} $.
The hint the professor gave us is to use generalized associative law. That is, Let $a_i \in$ $[0,\infty)$ for all $i\in I$ and let $(I_j)_{j\in J}$ be a disjoint family of subsets of I such that I =$\cup_{j \in J}I_j$, then $\sum_{j\in J}\sum_{i\in I_j}a_i=\sum_{i \in I}a_i$ .
Since I have already proved generalized associative law in previous question, I suppose I can use this result directly and to prove the question above I just write $w\times w=\cup_{n\in w}\cup_{i+j=w}${(i,j)}, which is disjoint family of subsets of $w\times w$, and then I just claim that the equation holds because they satisfy the condition of generalized associative law.