The product of countable set is countable

這有個很漂亮的證法, 運用質數的性質就不需要再實際造出函數了。Let f : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} be f(x, y) = 2^{x} 3^{y} which is a injective function mapping to \mathbb{N}. Hence there is a bijective function f' : \mathbb{N} \times \mathbb{N} \rightarrow A \subset \mathbb{N} where A is infinite. So that, \mathbb{N} \times \mathbb{N} is countable.

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