Open Real Interval is Subset of Closed Real Interval

Theorem
Let $a, b \in \R$ be real numbers.

Then:
 * $\left({a \,.\,.\, b}\right) \subseteq \left[{a \,.\,.\, b}\right]$

where:
 * $\left({a \,.\,.\, b}\right)$ is the open interval between $a$ and $b$
 * $\left[{a \,.\,.\, b}\right]$ is the closed interval between $a$ and $b$.

Proof
Let $x \in \left({a \,.\,.\, b}\right)$.

Then by definition of open interval:
 * $a < x < b$

Thus:
 * $a \le x \le b$

and so by definition of closed interval:
 * $x \in \left[{a \,.\,.\, b}\right]$.