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Homework Statement
Show that every infinite set contains a countably infinite set.
Homework Equations
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?