Proving Every Infinite Set Has An Infinite Countable Subset

[Doom of Doom]

So, the task is to prove: Every infinite set has a infinite countable subset.


2. A set $$S$$ is countable if there exists a bijection $$\phi: \mathbb{N}\rightarrow X$$

The Attempt at a Solution

 

[Pere Callahan]

You can easily construct a countable subset $\{s_1,s_2,\dots\}$ wiht $s_1,s_2,\dots$ being elements of S.

Let $s_1$ be some element of S. Let inductively $s_n$ be some element of $S-\{s_1,\dots,s_{n-1}\}$.

Can you show that this works for any infinite set, and that it does not work for any finite set?