Every countably infinite set has a countably infinite set

[k3k3]

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)=a_{n} for some a_{n}.

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

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

 

[Dick]

Yes, it is. I would rephrases some of the sentences, but in spirit it's correct.