Proving Every Infinite Set Has a Countable Subset

[Doom of Doom]

So, the task is to prove: Every infinite set has a infinite countable subset.2. A set X is countable if there exists a bijection \phi: \mathbb{N}\rightarrow X3. So here's what I have:

Let A be an infinite set, and pick some a_{1}\in A. Define S_{n}=\left\{a_{i}, i\in \mathbb{N} \left| 1\leq i \leq n \right\}.
Pick a_{n}\in (A - S_{n-1}) for each n \in \mathbb{N}, n>1.

Let X=\left\{a_{n}|n \in \mathbb{N}\right\}, and let \phi: \mathbb{N}\rightarrow X by \phi(x)=a_{x}.

Then I can show that \phi is a bijection, and thus I am done. Is this good? I'm sure there has to be a better way to do this.

 

[Pere Callahan]

I think it's alright. It's essentially what I suggested, but my answer has disappeared.. maybe it was too explicit..?

 

[Dick]

I think it's alright. It's essentially what I suggested, but my answer has disappeared.. maybe it was too explicit..?

Your post didn't disappear. It's on the other thread called 'countable sbset'. Don't repost the same question Doom of Doom.

 

[Pere Callahan]

Oh, I see, thanks Dick

 

[Doom of Doom]

I am sorry. My internet crashed right when i submitted the post, then when I went back later I didn't see my original post, so I made a new one.