Closure of Real Interval is Closed Real Interval/Proof 2/Lemma 1

Lemma for Closure of Real Interval is Closed Real Interval

Let $I$ be a non-empty real interval such that one of these holds:

$I = \openint a b$

$I = \hointr a b$

$I = \hointl a b$

$I = \closedint a b$

Let $I^-$ denote the closure of $I$.

Then:

$x \in \closedint a b \implies x \in I^-$

Proof

Let $x \in \closedint a b$.

Let $\openint c d$ be an open interval in $\R$ such that $x \in \openint c d$.

We must show that $\openint c d$ 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 $x \in \openint c d$.

Therefore $\openint c d$ contains a point in $I$.

$\Box$

Case: $x = a$

If $I$ contains $a$, then this means $x \in I$, and the proof is complete.

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 $r \le d$ and $r \le b$.

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

Thus, by Real Numbers are Densely Ordered, there exists some $s \in \R$ such that $a < s < r$.

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

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

Hence, $s$ is a point in $\openint c d$ which is also in $I$.

The existence of such a point is what we wanted to show.

$\Box$

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 $c \le l$ and $a \le l$.

$\blacksquare$