If $f:\mathbb{R}\to\mathbb{R}$ then $|f(\mathbb{N})|\leq\aleph_0$

Question

I am stuck at this problem for long time:

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Prove that if $f:\mathbb{R}\to\mathbb{R}$ is some real-valued function, Then $|f(\mathbb{N})|\leq\aleph_0$.

In other words, prove that there exists a one-to-one function from $f(\mathbb{N})=\{f(n)|n\in\mathbb{N}\}$ to $\mathbb{N}$.

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If $f(\mathbb{N})$ is finite then it is clear that $|f(\mathbb{N})|\leq\aleph_0$, But if $f(\mathbb{N})$ is infinite then I got stuck.

Thanks for any hint/help.

Answer 1

HINT: $\vert S \vert \le \aleph_0$ is equivalent to, by definition, the existence of a function with $f(\mathbb{N}) = S$.