Closure of Real Interval is Closed Real Interval

Theorem
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 $I^-$ is the closed real interval $\closedint a b$.