Closure of Real Interval is Closed Real Interval

Theorem
Let $I$ be a non-empty real interval such that one of these holds:
 * $I = \left({a \,.\,.\, b}\right)$
 * $I = \left[{a \,.\,.\, b}\right)$
 * $I = \left({a \,.\,.\, b}\right]$

or
 * $I = \left[{a \,.\,.\, b}\right]$

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

Then $I^-$ is the closed real interval $\left[{a \,.\,.\, b}\right]$.