# I need help with proofs pertaining to countability [duplicate]

Let $A$ and $B$ be countable sets.

(a) Show that $A \times B$ is countable. Hint: Show that there is a bijection from $A\times B$ onto a subset of $\Bbb Z \times\Bbb Z$:

(b) Use induction on $n$ to show that $A_1 \times A_2 \times \ldots \times A_n$ is countable if $A_1, A_2,\ldots, A_n$ are countable.

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## marked as duplicate by Asaf Karagila, Amr, JSchlather, TMM, QiL Dec 14 '12 at 22:39

Have you tried searching the site? – Asaf Karagila Dec 14 '12 at 21:04
Was the hint not helpful enough? – Hagen von Eitzen Dec 14 '12 at 21:08
Note, though that the proofs in the answers to the question that @Amr mentions are very different from the one suggested in your hint. They are direct proofs; yours makes use of the result that $\Bbb Z\times\Bbb Z$ is countable, which presumably you’ve already proved. – Brian M. Scott Dec 14 '12 at 21:19
Note that the question found by @Asaf covers only (b). – Brian M. Scott Dec 14 '12 at 21:21
@Brian: Thanks, luckily Amr's duplicate covers (a). – Asaf Karagila Dec 14 '12 at 21:26

A small additional nudge for (a): if $f:W\to Y$ and $g:X\to Z$ are injections, the map
$$W\times X\to Y\times Z:\langle w,x\rangle\mapsto\big\langle f(w),g(x)\big\rangle$$