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:


 * $\inf \left( A \right) \in \operatorname{cl} \left( A \right)$

From Closed Set equals its Topological Closure:


 * $A = \operatorname{cl} \left( A \right)$

Therefore $\inf \left( A \right) \in A$.