Closed Ordinal Space is Compact

Theorem
Let $\Gamma$ be a limit ordinal.

Let $\left[{0 \,.\,.\, \Gamma}\right]$ denote the closed ordinal space on $\Gamma$.

Then $\left[{0 \,.\,.\, \Gamma}\right]$ is a compact space.

Proof
By definition, $\left[{0 \,.\,.\, \Gamma}\right]$ is a linearly ordered space.

The result follows from Linearly Ordered Space is Compact iff Complete.