Proving that a subset of a countable set is countable

[Mathmellow]

I am trying to prove that any subset of a countable set is either finite or countable.

I know that a set \(\displaystyle S\) is countable if there exists a bijection that takes S to \(\displaystyle \Bbb{N}\). My first thought was to consider the subset \(\displaystyle V\) of \(\displaystyle S\). If \(\displaystyle V\) is finite we are done, since we can always construct a finite subset of a countably infinite set.
So I guess in the case where \(\displaystyle V\) is infinite we want to prove that there is a bijection \(\displaystyle \beta: V\to\Bbb{N}\). However, I am not sure how to do this.

I would really appreciate it if someone could help me with this, so I can feel more comfortable with these types of problems in the future!

 

[caffeinemachine]

I am trying to prove that any subset of a countable set is either finite or countable.

I know that a set \(\displaystyle S\) is countable if there exists a bijection that takes S to \(\displaystyle \Bbb{N}\). My first thought was to consider the subset \(\displaystyle V\) of \(\displaystyle S\). If \(\displaystyle V\) is finite we are done, since we can always construct a finite subset of a countably infinite set.
So I guess in the case where \(\displaystyle V\) is infinite we want to prove that there is a bijection \(\displaystyle \beta: V\to\Bbb{N}\). However, I am not sure how to do this.

I would really appreciate it if someone could help me with this, so I can feel more comfortable with these types of problems in the future!

One just has to show that any infinite subset of $\mathbb N$ is in bijection with $\mathbb N$.

Let $A\subseteq \mathbb N$ be infinite. Define a map $f:\mathbb N\to A$ by declaring $f(i)$ to be the $i$-th smallest element of $A$. Then $f$ is a bijection.