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If $A$ and $B$ are countable, then $A\times B$ is countable.

My attempt:

Lemma 1: $A$ is countable if and only if there exists a injective mapping $f:A \to \Bbb N$.

Lemma 2: $\Bbb N\times\Bbb N$ is countable.

Since $A$ and $B$ are countable, there exist injections $j_A:A \to \Bbb N$ and $j_B:B \to \Bbb N$ by Lemma 1.

We define a mapping $j:A\times B \to \Bbb N\times\Bbb N$ by $j(a,b)=(j_A(a),j_B(b))$. It's clear that $j$ is injective.

Since $\Bbb N\times\Bbb N$ is countable by Lemma 2, there exists an injection $f:\Bbb N\times\Bbb N \to \Bbb N$ by Lemma 1.

Hence $f\circ j:A\times B \to \Bbb N$ in injective and hence $A\times B$ is countable by Lemma 1.

Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help!

Akira
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    "It's clear that $j$ is injective" is not enough, you still need a proof, but it is good apart from this – ℋolo Sep 12 '18 at 17:08
  • Do you know that a countable union of countable sets is countable? That's another route. – Randall Sep 12 '18 at 17:10
  • @Randall I have just presented a proof for it at https://math.stackexchange.com/questions/2914550/if-a-is-a-countable-set-and-each-a-in-a-is-countable-then-bigcup-a-in ^^ – Akira Sep 12 '18 at 17:11
  • You have already solved a generalization of this problem here:https://math.stackexchange.com/questions/2914514/the-cartesian-product-of-a-finite-number-of-countable-sets-is-countable Edit Re: the comments below, my mistake! – Mike Earnest Sep 12 '18 at 17:12
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    @LeAnhDung Great! Then note that $A \times B = \bigcup_{a \in A} {a} \times B$. – Randall Sep 12 '18 at 17:12
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    @MikeEarnest he use this to prove the generalization – ℋolo Sep 12 '18 at 17:14
  • @MikeEarnest Actually no :) The theorem in this post is actually a lemma in the post you linked. – Akira Sep 12 '18 at 17:14
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    @Holo I just meant that. You are faster than me ^^ – Akira Sep 12 '18 at 17:14
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    @LeAnhDung hah, nice(after sitting on my butt and being on the computer most of the day I learned to type fast :P) – ℋolo Sep 12 '18 at 17:16
  • @Randall your approach is much more elegant if we've already known some lemmas ^^ – Akira Sep 12 '18 at 17:17
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    For some entertaining and informative reading on this subject I suggest "Stories About Sets" by Vilenkin. – DanielWainfleet Sep 13 '18 at 04:40
  • Thank you @DanielWainfleet! Let me have a look at it. – Akira Sep 13 '18 at 04:41

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