Let $X$ be a locally compact Hausdorff space such that $C_c(X),$ the space of all continuous functions with compact support is complete. Show that $X$ is compact.
I have shown that $C_c(X)$ is dense in $C_0(X),$ the space of all continuous functions vanishing at infinity. Since $C_c(X)$ is complete, therefore $$C_c(X)=C_0(X).$$
Now to conclude that $X$ is compact, it suffices to find a function in $C_0(X)$ which vanishes nowhere on $X.$ Is this always possible?