Closed Subset of Real Numbers with Lower Bound contains Infimum

Theorem
Consider the real number line as a metric space under the usual metric.

Let $A \subseteq \R$ such that $A$ is closed in $\R$ and $A \neq \varnothing$.

Let $A$ be bounded below.

Then $A$ contains its infimum.

Proof
From Infimum of Bounded Below Set of Reals is in Closure:


 * $\map \inf A \in \map \cl A$

From Set is Closed iff Equals Topological Closure:


 * $A = \map \cl A$

Therefore $\map \inf A \in A$.