Prove $A$ is either finite or countable.

Question

Let $A$ be a set and let $f:A \to \Bbb{N}$ be an injection. Prove $A$ is either finite or countable. How can one prove such a thing? Since this is so obvious it gets me in a place where I don't know what I can and cannot assume. I would appreciate your help.

Answer 1

Since $f$ is an injection, you know that the map $f\colon A\rightarrow f(A)\subseteq\mathbb{N}$ is a bijection. This means that $A$ and $f(A)$ have the same cardinality. You are then left with the problem of considering the possible cardinalities of a subset $f(A)$ of $\mathbb{N}$. I hope this helps.