Proof that an infinite subset of $\mathbb{N}$ is countable

I'd rather use the well known and comfortable property of the natural numbers of being a well ordered set:

First, let $\;a_1\in A\;$ be the smallest (wrt the usual order) element in $\;A\;$

Now, let $\;a_2\in A\setminus\{a_1\}\;$ be the smallest element, etc.