Could you help me prove the following fact? I've been trying to prove it and I've searched for a hint in Englking's book, but I haven't come up with anything:
If $X$ is a Hausdorff space and each $x \in X$ has at least one compact neighbourhood, then $X$ is locally compact.
$U$ is a neighbourhood of $x \in X$ $\iff$ $\exists V$ - open $: x \in V \subset U$.
$X$ is locally compact $\iff$ each $x \in X$ has a basis of compact neighbourhoods.