(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that every infinite set contains a countably infinite set.

2. Relevant equations

3. The attempt at a solution

Let A be an infinite set

Since A is infinite and not empty, there exists an a∈A

Let f be a mapping such that for all n in ℕ such that f(n)=[itex]a_{n}[/itex] for some [itex]a_{n}[/itex].

Then f(1)=[itex]a_{1}[/itex]. A is an infinite set, so there exists another element [itex]a_{2}[/itex] that is not [itex]a_{1}[/itex] such that f(2)=[itex]a_{2}[/itex] and so on.

This makes f a 1-1 function.

Since every element of ℕhas an image, the subset generated by f is onto ℕ.

Therefore, since f creates a subset of A, infinite subsets have a countable subset.

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Is this a valid proof on why infinite sets have a countably infinite subset?

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# Every countably infinite set has a countably infinite set

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