My problem reads:
Prove that if there is a function $f\colon A\rightarrow \mathbb{N}$ that is one-to-one, then $A$ is countable
Assuming $\mathbb{N}$ to be the set of natural numbers.
I am not too sure how to go about proving this. Would I need the definition of denumerable in this case? or can I use the cardinality of A < or equal to N?
Best Answer
If $f\colon A\rightarrow \mathbb{N}$ is one-to-one, then $f\colon A\rightarrow f(A)$ is bijective. Now, since $f(A)\subseteq \mathbb{N}$, then by this result $f(A)$ is countable, which gives us that $A$ is countable too.