Mathmellow
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I am trying to prove that any subset of a countable set is either finite or countable.
I know that a set $$S$$ is countable if there exists a bijection that takes S to $$\Bbb{N}$$. My first thought was to consider the subset $$V$$ of $$S$$. If $$V$$ 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 $$V$$ is infinite we want to prove that there is a bijection $$\beta: V\to\Bbb{N}$$. 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!
I know that a set $$S$$ is countable if there exists a bijection that takes S to $$\Bbb{N}$$. My first thought was to consider the subset $$V$$ of $$S$$. If $$V$$ 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 $$V$$ is infinite we want to prove that there is a bijection $$\beta: V\to\Bbb{N}$$. 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!