# What is the name of the technique for showing that $\mathbb{N}^2$ is countable?

In order to show that $\mathbb{N} \times \mathbb{N}$ is countable, we can define a bijection $f : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ like this one:

• $0 \rightarrow (0, 0)$
• $1 \rightarrow (1, 0)$
• $2 \rightarrow (0, 1)$
• $3 \rightarrow (2, 0)$
• $4 \rightarrow (1, 1)$
• $5 \rightarrow (0, 2)$
• $6 \rightarrow (3, 0)$
• $7 \rightarrow (2, 1)$
• $8 \rightarrow (1, 2)$
• ...

I need to prove that a set is countable, I know how to use this proof method, but I can't remember its name. Any hints?

• You are just establishing a bijection between $\mathbb{N}\times \mathbb{N}$ and $\mathbb{N}$. You can call this enumeration of $\mathbb{N}\times \mathbb{N}$. – baharampuri May 16 '15 at 10:04
• @ChristianBlatter: fixed that – hey hey May 16 '15 at 14:27
• @baharampuri: in my context this is only one of a few possible enumerations. This is why I needed a name for it. – hey hey May 16 '15 at 14:28

Aha, I found it! The $f$ function I'm using is Cantor's pairing function.