Discrete Space is Paracompact

Theorem
Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Then $T$ is paracompact.

Proof
Let $\mathcal V$ be any open cover of $S$.

Consider the set $\mathcal C$ of all singleton subsets of $S$:
 * $\mathcal C := \left\{{\left\{{x}\right\}: x \in S}\right\}$

From Discrete Space has Open Locally Finite Cover, $\mathcal C$ is an open cover which is locally finite.

This result also shows that $\mathcal C$ is the finest cover on $T$.

So $\mathcal C$ is an open refinement of $\mathcal V$ which is locally finite.

So $T$ is paracompact, by definition.