Closed Real Interval is Closed Set

Theorem
Let $\R$ be the real number line considered as an Euclidean space.

Let $\closedint a b \subset \R$ be a closed interval of $\R$.

Then $\closedint a b$ is a closed set of $\R$.

Proof
From the corollary to Open Real Interval is Open Set, both $\openint {-\infty} a$ and $\openint b \infty$ are open sets in $M$.

From Union of Open Sets of Metric Space is Open it follows that $\openint {-\infty} a \cup \openint b \infty$ is open in $\R$.

But $\openint {-\infty} a \cup \openint b \infty$ is the relative complement of $\closedint a b$ in $\R$.

The result follows by definition of closed set.