Heine-Borel Theorem/Real Line/Closed and Bounded Set

Theorem

Let $F$ be a closed and bounded real set.

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

Let $C$ be a cover of $F$.

Then there is a finite subset of $C$ that covers $F$.

Proof

We are given that $C$ is a set of open real sets that covers $F$.

In other words, $C$ is an open cover of $F$.

We need to show that there is a finite subset of $C$ that covers $F$.

In other words, we need to show that $C$ has a finite subcover.

Let $F_o$ be the complement of $F$ in $\R$.

By definition of closed real set, $F_o$ is open as $F$ is closed.

Step 1: $C^*$ is an Open Cover of $\closedint a b$

It is demonstrated that $C^*$ is an open cover of $\closedint a b$.

Since $F$ is bounded, $F$ is contained in a closed and bounded interval $\closedint a b$ where $a, b \in \R$.

Define $C^* = C \cup \set {F_o}$.

Like $C$, $C^*$ is a set of open real sets as $F_o$ is open.

$C^*$ covers $F \cup F_o$ as $C$ covers $F$ and $\set {F_o}$ covers $F_o$.

$F_o \cup F$ equals $\R$ as $F_o$ is the complement of $F$ in $\R$.

So $C^*$ covers $\R$.

Furthermore, $C^*$ is an open cover of $\closedint a b$ as $\closedint a b$ is a subset of $\R$.

Step 2: $C^*$ has a Finite Subcover

It is demonstrated that $C^*$ has a finite subcover $C^*_f$.

$C^*$ is an open cover of the closed and bounded interval $\closedint a b$.

Therefore, by Open Cover of Closed and Bounded Real Interval has Finite Subcover, $C^*$ has a finite subcover $C^*_f$.

Step 3: $C$ has a Finite Subcover

It is demonstrated that $C$ has a finite subcover $C_f$.

Note that $F_o$ is the only element in $C^*$ that is not an element of $C$.

Therefore, $F_o$ is the only possible element in $C^*_f$ that is not an element of $C$ as $C^*_f$ is a subset of $C^*$.

This means that $C^*_f \setminus \set {F_o}$ is a subset of $C$.

Define $C_f = C^*_f \setminus \set {F_o}$.

According to the reasoning above, $C_f \subseteq C$.

Also, $C_f$ is finite as $C^*_f$ is finite.

What remains is to show that $C_f$ covers $F$.

We have:

$\ds F$

$\subseteq$

$\ds \closedint a b$

$\, \ds \land \, $

$\ds \closedint a b$

$\subseteq$

$\ds \bigcup_{O \mathop \in C^*_f} O$

as $C^*_f$ is a cover of $\closedint a b$

$\ds \leadsto \ \ $

$\ds F$

$\subseteq$

$\ds \bigcup_{O \mathop \in C^*_f} O$

Subset Relation is Transitive

$\ds \leadsto \ \ $

$\ds F \cap \relcomp \R {F_o}$

$\subseteq$