Existence of a non-zero continuous function vanishing at infinity.

Question

I am currently reading the book '$C^*$-Algebras and Operator Theory' by Gerard J. Murphy and I have trouble understanding two statements on page 4.

Let $\Omega$ be a locally compact Hausdorff space and $a \in \Omega$ be a fixed point. Denote by $C_0(\Omega)$ the set of continuous functions $f: \Omega \to \mathbf{C}$ that vanish at infinity, i.e. for every $\varepsilon > 0$ the set {$\omega \in \Omega: |f(\omega)| \geq \varepsilon$} is compact.

Now, the claim is that there exists an $u \in C_0(\Omega)$ s.t. $u(a) = 1$. I am not sure how one would prove this statement. I was thinking about the Hahn-Banach theorem, but I can't bend this situation into one where I get a sensible result using this theorem.

Answer 1

Let $K$ be a compact neighbourhood of $a$. It contains some open neighborhood $U$ of $a$. As a locally compact Hausdorff space, $\Omega$ is completely regular hence there exists a continuous function $f:\Omega\to\Bbb R$ such that $f(a)=1$ and $f=0$ outside $U$, and a fortiori outside $K$.

Answer 2

Let $W$ be an open neighbourhood of $a$ in $\Omega$ such that $\overline{W}$ is compact. As a compact Hausdorff space, $\overline{W}$ is normal. Use Urysohn's Lemma to get a continuous function $f$ on $\overline{W}$ such that $f(a) = 1$ and $f = 0$ on $\overline{W} \backslash W$. Extend to be $0$ on $\Omega \backslash W$.