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?