Cover is Cover of Subset

Theorem

Let $S$ be a set.

Let $\CC$ be a cover of $S$.

Let $T \subseteq S$ be a subset of $S$.

Then, $\CC$ is a cover of $T$.

Proof

By definition of a cover:

$\ds S \subseteq \bigcup C$

But then, by Subset Relation is Transitive:

$\ds T \subseteq \bigcup C$

Therefore, $C$ is a cover of $T$ by definition.

$\blacksquare$