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

### Step 1: $C^*$ is an Open Cover of $\left[{a \,.\,.\, b}\right]$

It is demonstrated that $C^*$ is an open cover of $\left[{a \,.\,.\, b}\right]$.

Since $F$ is bounded, $F$ is contained in a closed and bounded interval $\left[{a \,.\,.\, b}\right]$ where $a, b \in \R$.

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

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 $\left\{ {F_o} \right\}$ 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 $\left[{a \,.\,.\, b}\right]$ as $\left[{a \,.\,.\, b}\right]$ 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 $\left[{a \,.\,.\, b}\right]$.

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 $C^*_f$ \ $\left\{ {F_o} \right\}$ is a subset of $C$.

Define $C_f$ = $C^*_f$ \ $\left\{ {F_o} \right\}$.

According to the reasoning above, $C_f$ is a subset of $C$.

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

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

We have:

\(\ds F \subseteq \left[ {a \,.\,.\, b} \right]\) | \(\implies\) | \(\ds F \subseteq \left[ {a \,.\,.\, b} \right] \subseteq \bigcup_{O \mathop \in C^*_f} O\) | as $C^*_f$ is a cover of $\left[ {a \,.\,.\, b} \right]$ | |||||||||||

\(\ds \) | \(\implies\) | \(\ds F \subseteq \bigcup_{O \mathop \in C^*_f} O\) | ||||||||||||

\(\ds \) | \(\implies\) | \(\ds F \cap \complement F_o \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) | Set Intersection Preserves Subsets | |||||||||||

\(\ds \) | \(=\) | \(\ds F \cap \complement \complement F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) | as $F_o = \complement F$, the complement of $F$ in $\R$ | |||||||||||

\(\ds \) | \(=\) | \(\ds F \cap F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) | Relative Complement of Relative Complement | |||||||||||

\(\ds \) | \(=\) | \(\ds F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) | Intersection is Idempotent |

Furthermore, as $F \subseteq \left[{a \,.\,.\, b}\right]$:

\(\ds F\) | \(\subseteq\) | \(\ds \left({\bigcup_{O \mathop \in C^*_f} O}\right) \cap \complement F_o\) | ||||||||||||

\(\ds \) | \(\subseteq\) | \(\ds \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right) \cap \complement F_o\) | by $\displaystyle \left({\bigcup_{O \mathop \in C^*_f} O}\right) \subseteq \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right)$ and Set Intersection Preserves Subsets | |||||||||||

\(\ds \) | \(=\) | \(\ds \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right) \setminus F_o\) | as set intersection with complement equals set difference | |||||||||||

\(\ds \) | \(=\) | \(\ds \left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} \cup \left\{ {F_o}\right\} }\right)} O}\right) \setminus F_o\) | by Set Difference Union Second Set is Union | |||||||||||

\(\ds \) | \(=\) | \(\ds \left({\left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O}\right) \cup F_o}\right) \setminus F_o\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O}\right) \setminus F_o\) | Set Difference with Union is Set Difference | |||||||||||

\(\ds \) | \(\subseteq\) | \(\ds \bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O\) | Set Difference is Subset | |||||||||||

\(\ds \) | \(=\) | \(\ds \left({\bigcup_{O \mathop \in C_f} O}\right)\) | Definition of $C_f$ |

Thus, $C_f$ covers $F$.

This finishes the proof.

$\blacksquare$

## Source of Name

This entry was named for Heinrich Eduard Heine and Émile Borel.

## Sources

- 1988: H.L. Royden:
*Real Analysis*(3rd ed.): Ch. $2$: Section $5$, theorem $15$