Talk:Cartesian Product of Countable Sets is Countable
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Can't we formalize the informal proof? To count $\mathbb N^2$, we have
$(0,0)$
$(0,1),(1,0)$
$(0,2),(1,1),(2,0)$
etc. We can define this recursively:
$s_1 = (0,0)$
If $(s_n)_2 = 0$, then $s_{n+1} = (0,(s_n)_1+1)$.
If $(s_n)_2 \ne 0$, then $s_{n+1} = ((s_n)_1+1, (s_n)_2-1)$.
Now $(s_n)_1 + (s_n)_2$ is monotonically increasing, so it eventually reaches any given natural. It's then clear that $s_n$ reaches any ordered pair of naturals. I'll try to do this sometime soon.--Dfeuer (talk) 03:53, 21 December 2012 (UTC)
- As long as you don't replace that existing informal proof, but instead create a new instance of a proof, knock yourself out. --prime mover (talk) 10:28, 21 December 2012 (UTC)