Existence of Paracompact Space which is not Compact

Theorem
There exists at least one example of a paracompact topological space which is not also a compact topological space.

Proof
Let $T$ be the real number line under the Euclidean topology.

From Real Number Space is Paracompact, $T$ is a paracompact space.

From Real Number Space is not Countably Compact, $T$ is not a countably compact space.

From Compact Space is Countably Compact, it follows that $T$ is not a compact space.

Hence the result.