Lindelöf's Lemma

Theorem
Let $C$ be a set of open real sets.

Let $S$ be a real set that is covered by $C$.

Then there exists a countable subset of $C$ that covers $S$.

Lemma
We have that $S$ is covered by $C$.

This means that $S$ is a subset of $\displaystyle \bigcup_{O \mathop \in C} O$.

From the lemma:
 * $\displaystyle \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$

where $D$ is a countable subset of $C$.

Hence $S$ is also a subset of $\displaystyle \bigcup_{O \mathop \in D} O$.

In other words, $S$ is covered by $D$.

That is, $S$ is covered by a countable subset of $C$.

This finishes the proof of the theorem.