Real Number Line is not Countably Compact

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

Then $\left({\R, \tau_d}\right)$ is not countably compact.

Proof
Let $\mathcal C$ be the set of subsets of $\R$ defined as:
 * $\mathcal C = \left\{{\left({n \,.\,.\, n + 2}\right): n \in \Z}\right\}$

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

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

Hence the result, by definition of countably compact.