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**Problem:**

Suppose that $\left\{ X_n \right\}_{n=1}^{\infty}$ is a sequence of compact, Hausdorff spaces and for each $n, f_n : X_{n+1} \rightarrow X_n$ is a continuous function (not necessarily onto).

Show that:

$X = lim_{\leftarrow} \left\{ X_n, f_n \right\}_{n=1}^{\infty} \neq \emptyset$

Furthermore, show that $X$ is compact.

I have seen a proof for a general inverse limit system, with $D$ being its directed set. However, I assume the proof differs in the problem I've stated. (here I guess our $D = \Bbb{N}$).

Does anyone know of a proof for this online?

Or perhaps can give an outline for the proof?