[Math] Show that the set of isolated points of $S$ is countable

Question

Let $S$ be a subset of $\mathbb{R}^n$; show that the set $I$ of isolated points of $S$ is countable.

Let $\mathbf{x}\in I$. There exists an open ball, say $B(\mathbf{x},r_\mathbf{x})$, of radius $r_\mathbf{x}$, for each $\mathbf{x}$ such that $B(\mathbf{x},r_\mathbf{x}) \cap S=${$\mathbf{x}$}. Now without change of notation replace $ r_\mathbf{x}$ with $\frac{1}{2}r_\mathbf{x}$. Thus the open balls $B$ are pairwise disjoint. We must define an injective function $f:I\to \mathbb{N}$. Which one should I choose?

And…any idea for a second proof?

Answer 1

Hint Any ball contains a point with all coordinates rational, and $\mathbb Q^n$ is countable.