Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative upper semicontinuous function $f$ on $\beta X$ such that $f$ is positive on $X $ and $f\left( p\right) =0$, then $X$ is realcompact.

I looked at both of John Mack's articles in the references but couldn't find this result. How can I prove this result?