# Proof of Fubini's Theorem for Series

I'm currently trying to self study Tao's Analysis I, and I'm really struggling with this proof - as I feel like I'm missing something. He goes really slow over sections I feel are obvious - for example he spends a while trying to get me to justify that for: $$f(x, y) \geq 0, \sum_{(x, y) \in A} f(x, y) \leq \sum_{(x, y) \in \mathbb{N} \times \mathbb{N}} f(x, y),$$ for $$A \subset \mathbb{N} \times \mathbb{N}$$ which seems intuitive to me. He suggests using a bijection to map it to $$\mathbb{N}$$ and I don't see why that's necessary. Therefore, I feel like I'm missing something critical in this section. I just started this book again after not reading it for a couple months - could this be why?

• The summand in the second sum is missing. Is it supposed to be $f(x,y)$? – saulspatz Jul 21 at 15:50
• @evaristegd thanks! – It'sNotALie. Jul 21 at 15:55
• @saulspatz yes, thank you :) – It'sNotALie. Jul 21 at 15:55