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I am trying to show that there exists a bijective function $f:\mathbb{N}\rightarrow \mathbb{N} \times \mathbb{N}$ without using prime factorization. I have pieces here and there, but unable to put togther a complete proof. It would be great someone could help me out.

This is what I roughly got. I choose a sequence of postive integer $a_n=\frac{n(n+1)}{2}$ and I know they will be "captured once" by the interval $I_n=(a_{n-1},a_n]$ for all $n\geq 2$. I will let $I_1=\{1\}$. Then once show that there is indeed a bijective function between the set of intervals and the set of number defined by the sequence. I will invoke the property that any infintie subset of a countable set is countable to complete the proof.

chuck
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    Look up the Cantor pairing function. It occurs here often. – Ross Millikan Sep 19 '17 at 20:49
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    Use the first diagonal theorem of Cantor. He uses that to show that $\mathbb{Q}$ is countable, but you can use it also to show that $\mathbb{N}\times\mathbb{N}$ is countable. – S. M. Roch Sep 19 '17 at 20:50

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