Indiscrete Space is Hereditarily Compact

Theorem
Let $(X,\tau)$ be an indiscrete topological space.

Then $(X,\tau)$ is hereditarily compact.

Proof
By Subsets of Indiscrete Space are Compact and Sequentially Compact, we find that $(X,\tau)$ is hereditarily compact.

Also see

 * Hausdorff Space is Hereditarily Compact Iff Finite Discrete Space