Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact

Theorem
Let $I$ be an indexing set with uncountable cardinality.

Let $\family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$.

Let each of $\struct{S_\alpha, \tau_\alpha}$ be sequentially compact.

Then it is not necessarily the case that $\struct{S, \tau}$ is also sequentially compact.

Proof
Let $\mathbb I$ denote the closed unit interval.

Let $T = \struct{\mathbb I, \tau}$ be the topological space consisting of $\mathbb I$ under the usual (Euclidean) topology.

Let $T' = \mathbb I^{\mathbb I} = \struct{\displaystyle \prod_{\alpha \mathop \in \mathbb I} \struct{\mathbb I, \tau}_\alpha, \tau'}$ be the uncountable Cartesian product of $\struct{\mathbb I, \tau}$ indexed by $\mathbb I$ with the Tychonoff topology $\tau'$.

From Closed Real Interval is Sequentially Compact, $T$ is sequentially compact.

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

Hence the result.