Existence of Weakly Countably Compact Space which is not Countably Compact

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

Proof
Let $T$ be the deleted integer topology.

From Deleted Integer Topology is Weakly Countably Compact, $T$ is a weakly countably compact space.

From Deleted Integer Topology is not Countably Compact, $T$ is not a countably compact space.

Hence the result.