Inductive limit of compact Hausdorff spaces being realcompact

Question

Suppose that $\{X_n\}_{n\in \mathbb{N}}$ is an increasing sequence of compact Hausdorff spaces and $X$ is their union, equipped with the finest topology under which all the inclusion maps are continuous. Is it well-known that $X$ is realcompact? Is there a reference or an easy argument for this fact? Thank you very much.

Answer 1

Clearly your $X$ is Lindelöf, so it follows from Theorem 8.2 in Gillman & Jerison, Rings of Continuous Functions: Every Lindelöf space is realcompact.