Real Number Line is Paracompact

Theorem
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Then $\struct {\R, \tau_d}$ is paracompact.

Proof
Let $\CC$ be an open cover for $\R$.

Then $\CC$ covers each of the closed real intervals $\closedint n {n + 1}$ for all $n \in \Z$.

By the Heine-Borel Theorem, each of $\closedint n {n + 1}$ is compact.

So, for each of these intervals $\closedint n {n + 1}$, it follows that $\CC$ can be reduced to a sequence $\sequence {G_i^{\paren n} }$ of finite subcovers.

Then each of $G_i^{\paren n} \cap \openint {n - 1} {n + 2}$ forms a refinement of $\CC$ which is locally finite.

Hence, by definition, $\R$ is paracompact.