Compact Space is Weakly Locally Compact/Proof 1

Proof
Let $T$ be a compact space.

By definition, $T$ is weakly locally compact each point of $T$ has a compact neighborhood in $T$.

Let $x \in S$ be an arbitrary point of $T$.

By Space is Neighborhood of all its Points, $T$ is a neighborhood of $x$.

Thus $x$ has a compact neighborhood in $T$.

Hence the result by definition of weakly locally compact.