Existence of Compact Space which is not Sequentially Compact

Theorem
There exists at least one example of a compact topological space which is not also a sequentially compact space.

Proof
Let $T = \mathbb I^{\mathbb I}$ be the uncountable Cartesian product of the closed unit interval under the Euclidean topology.

From Uncountable Cartesian Product of Closed Unit Interval is Compact, $T$ is a compact space.

From Uncountable Cartesian Product of Closed Unit Interval is not Sequentially Compact, $T$ is not a sequentially compact space.

Hence the result.