Subset of Real Numbers is Path-Connected iff Interval

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

Let $S \subseteq \R$ be a subset of $\R$.

Then $S$ is a path-connected metric subspace of $\R$ iff $S$ is a real interval.

Proof
Let $S$ be a subset of $\R$.

Necessary Condition
Let $S$ be a path-connected metric subspace of $\R$.

From Path-Connected Space is Connected, it follows that $S$ is connected.

From Only Intervals are Connected, it follows that $S$ is an real interval.

Sufficient Condition
Let $S$ be a real interval.