Existence of Weakly Locally Compact Space which is not Strongly Locally Compact/Proof 1

Proof
Let $T$ be the nested interval topological space.

From Nested Interval Topology is Weakly Locally Compact, $T$ is a weakly locally compact space.

From Nested Interval Topology is not Strongly Locally Compact, $T$ is not a strongly locally compact space.

Hence the result.