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Let $A$ be a countable set, and let $f : A \to B$ be a function. Prove that $f(A)$ is at most countable.

Therefore $A$ is bijective with positive integers.

Since the function is defined, it is clear that every elements in $A$ will not map to a different elements in $B$, i.e., for every $a\in A$ there exists a unique $b\in B$. So we don't care much whether the function is bijective or whatever.

I also know that $f(A)$ is at most countable if it is either finite or countable.

I think that $f(A)$ if proved as finite, should be a key to this proof, and to show if it is finite we may need to show something on the line that $|f(A)| < |A|$.

I don't know how all these pieces fit together to construct a proof. Please help!

Thank you.

Andrés E. Caicedo
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1 Answers1

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Here are some hints.

You can assume without loss of generality, that $A=\mathbb{N}$ and $f(A)=B$. (Details left to you.)

To show that $B$ is at most countable, it suffices to construct an injective function from $B$ to $\mathbb{N}$. Define $g: B\to \mathbb{N}$ such that $g(x)=\min\{n\in\mathbb{N}:f(n)=x\}$. Show that this is a well-defined injective function.

halrankard2
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