To prove that $X_{\infty}$ is normal

Question

If $((X_n),(f_n))$ is a inverse limit system, and for each $n$, $X_n\neq {\emptyset}$ is compact and Hausdorff then $X_{\infty}$ is normal.

I know there is a theory that says: If $((X_n),(f_n))$ an inverse limit system , and for each $n$, $X_n\neq{\emptyset}$ is compact and Hausdorff then $X_{\infty}$ is Hausdorff and compact. So by using another theory that every compact, Hausdorff space is normal , can it be solved?

Answer 1

The inverse limit of a system of (non-empty) compact Hausdorff spaces is again compact Hausdorff, hence normal. Yes, those two steps will do it.