Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set/Lemma 1

Lemma
Let $\struct {X, \tau}$ be a compact Hausdorff space.

Let $f : X \to X$ be a continuous function.

Define a sequence of sets $\sequence {X_i}_{i \in \N}$ by:


 * $X_i = \begin{cases}X & i = 1 \\ \map f {X_{i - 1} } & i \ge 2\end{cases}$

Then:
 * for each $i \in \N$, the set $X_i$ is non-empty and closed.

Proof
For all $n \in \N$, let $\map P n$ be the proposition:


 * the set $X_n$ is non-empty and closed.