Real Number Line is not Countably Compact

Theorem
Let $\struct {\R, \tau_d}$ be the real number line considered as a topological space under the usual (Euclidean) topology.

Then $\struct {\R, \tau_d}$ is not countably compact.

Proof
Let $\CC$ be the set of subsets of $\R$ defined as:
 * $\CC = \set {\openint n {n + 2}: n \in \Z}$

Then $\CC$ is an open cover of $\R$ which is countable.

However, there is no finite subcover for $\R$ of $\CC$.

Hence the result, by definition of countably compact.