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  "body": "\n## Statement\n\nLet $A$ and $B$ be [[countable sets|0ZG61TFBES]].\nThen the [[Cartesian product|0C81YBVTWH]] $A \\times B$ is countable.\n\n## Remarks\n\nProof: take injections $f \\colon A \\to \\mathbb{N}$ and $g \\colon B \\to \\mathbb{N}$. Then $(a, b) \\mapsto \\pi(f(a), g(b))$ is an injection to $\\mathbb{N}$, where $\\pi$ is the [[Cantor pairing function|0NP5N5PYW5]].\nBy induction, a finite product $A_1 \\times \\dots \\times A_n$ of countable sets is countable.\nAn infinite product need not be countable: the set of sequences of $0$ and $1$ is [[uncountable|0X55M9XET4]].\n\n## Explanations\n\n### L1 human\n\nThe pairs of natural numbers fill an infinite grid, and the grid can be listed diagonal by diagonal.\nSo a product of two lists is again a list.\n",
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