TITLE: Show that a locally compact Hausdorff space is regular.
QUESTION [7 upvotes]: Show that a locally compact Hausdorff space $(X,\tau)$ is regular.

I have already shown that a compact Hausdorff space is regular.
My textbook proposes 2 methodes, but I get stuck at both. The first method looks the most elegant, but how can I continue?
Compactification method.
Consider the compactification $(X_\infty, \tau_\infty)$ where $X_\infty = X\sqcup \infty$ and $\tau_\infty = \tau \cup \{ X_\infty \setminus K: K \text{ is a compact subset of } X\}$
Then $(X_\infty, \tau_\infty)$ is regular.
Let $x\in X\setminus F, F\subseteq X$ closed in $\tau$. Then $x\in X_\infty$ and $F\sqcup \{ \infty\}$ is closed in $\tau_\infty$. By regularity:
$$(\exists U_\infty, V_\infty \in \tau_\infty)(x\in U_\infty, (F\sqcup \{ \infty\} ) \subseteq V_\infty, U_\infty\cap V\infty = \varnothing)$$
Since $\infty \in V_\infty$ then $U_\infty \in \tau$.
How can I prove $V_\infty \setminus \{ \infty\} \in \tau?$ Any pointers?
Direct method:
Consider $x\in X\setminus F, F\subseteq X$ closed in $\tau$.
We are on the lookout for $(U, V\in \tau)$ such that $U\cap V = \varnothing, x\in U, F\subseteq V$.
Since $x\in X$ there (exists $K\subseteq X$ compact)($K$ is a neighbourhood of $x$)
Consider the compact subspace $(K, \tau_K)$ of $(X,\tau)$ which is Hausdorff. Then $K\cap F$ is closed in $\tau_K$ which makes $(K,\tau_K)$ regular. And then $(\exists U_K, V_K \in \tau_K)(x\in U_K, (K\cap F)\subseteq V_K, U_K\cap V_K =  \varnothing)$.
But how can I expand this $V_K$, such that $F\subseteq V$?

REPLY [6 votes]: The second approach can be made clearer (for my tastes, which do not have to agree with yours) by using the English language more.
A simple reformulation is that you are looking for a closed neighbourhood of $x$ that does not meet $F$. Now $x$ has a compact neighbourhood $K$ and a closed neighbourhood $W_K$ in $K$ that does not meet $F$. Since $K$ is a neighbourhood of $x$, $W_K$ is also a neighbourhood of $x$ in $X$, and since $K$ is closed, $W_K$ is also closed in $X$.
(The closed neighbourhood $W_K$ can be chosen as the closure of $U_K$ in your formulation of the proof.)