Proving that if a set $A$ is infinite then necessarily $|A|\geq|\mathbb{N}|$

Question

A set $A$ is set to be infinite if it is not finite, i.e. if there exists no $n\in\mathbb{N}$ such that $|A|=n$, meaning there exists a bijection $A\leftrightarrow\{1,\dotsc,n\}$. How do I prove that for any such set there necessarily exists an injective function $f:\mathbb{N}\to A$, meaning there are no infinite cardinalities under countability?

Answer 1

Suppose $|A|<|\mathbb{N}|$. Then there is an injection $A\to \mathbb{N}$ onto a proper subset that has smaller cardinality than $\mathbb{N}$. However, all infinite subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$, so it follows that $A$ is finite.