Closure of Real Interval

Theorem

Let $I$ be a nonempty real interval such that:

• $I = \left({a .. b}\right)$
• $I = \left[{a .. b}\right)$
• $I = \left({a .. b}\right]$ or
• $I = \left[{a .. b}\right]$.

Then $\operatorname{cl} \left({I}\right)$, the closure of $I$, is the closed interval $\left[{a .. b}\right]$.

Proof

Let $I$ be one of the intervals as specified in the exposition.

Note that by Condition for Point being in Closure, $x \in \operatorname{cl} \left({I}\right)$ if and only if every open set in $\R$ containing $x$ contains a point in $I$.

Furthermore, since every open set in the reals is a union of open intervals, we also have that $x \in \operatorname{cl} \left({I}\right)$ if and only if every open interval containing $x$ contains a point in $I$.

We will make use of this equivalence throughout the proof.

Lemma: $x \in \left[{a .. b}\right] \implies x \in \operatorname{cl} \left({I}\right)$

Let $x \in \left[{a .. b}\right]$.

Let $\left({c .. d}\right)$ be an open interval in $\R$ such that $x \in \left({c .. d}\right)$.

We must show that $\left({c .. d}\right)$ contains a point in $I$.

One of the following three possibilities holds:

• $a < x < b$
• $x = a$
• $x = b$

Case: $a < x < b$

In this case, $x \in I$ and $x \in \left({c .. d}\right)$, so $\left({c .. d}\right)$ contains a point in $I$.

Case: $x = a$

If $I$ contains $a$, then we are done since this means $x \in I$. So, assume that $a \notin I$.

Since $I$ is nonempty but does not contain $a$, we must have $a < b$.

Let $r$ be the minimum of $d$ and $b$, so that $r \leq d$ and $r \leq b$.

Since $a = x < d$ by choice of $d$ and since $a < b$ by assumption, we must have $a < r$.

Thus, by Reals are Close Packed, there exists some $s \in \R$ such that $a < s < r$.

To summarize, we have $c < x=a < s < r$, where $r\leq d$ and $r \leq b$.

This means that $s$ satisfies both $c < s < d$ and $a < s < b$.

Hence, $s$ is a point in $\left({c .. d}\right)$ which is also in $I$. The existence of such a point is what we wanted to show.

Case: $x = b$

This case is analogous to case when $x = a$.

Here we instead let $l$ be the maximum of $c$ and $a$, and select an $s$ such that $l < s < x=b < d$, where $c \leq l$ and $a \leq l$.

$\Box$

Lemma: $x \notin \left[{a .. b}\right] \implies x \notin \operatorname{cl} \left({I}\right)$

Suppose $x \notin \left[{a .. b}\right]$. We must find an open interval containing $x$ which does not contain a point in $I$.

There are two possibilities:

• $x < a$ or
• $x > b$

Case: $x < a$

By Reals are Close Packed, there exists $r \in \mathbb{R}$ such that $x < r < a$.

Thus $\left({x-1 .. r}\right)$ is an open interval, all of whose elements are less than $a$, and hence not in $I$.

Case: $x > b$

This is similarly to the case when $x < a$.

Here instead we pick $r$ such that $b < r < x$, and consider the interval $\left({r .. x+1}\right)$

$\Box$

By the two lemmas proven above, $\left[{a .. b}\right] = \operatorname{cl} \left({I}\right)$.

$\blacksquare$