Product of Paracompact Spaces is not always Paracompact

Theorem
Let $I$ be an indexing set.

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 a paracompact space.

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

Proof
Let $T$ be the real number line with the right half-open interval topology.

Let $T' = T \times T$ be Sorgenfrey's half-open square topology.

From Right Half-Open Interval Topology is Paracompact Space, $T$ is a paracompact space.

From Sorgenfrey's Half-Open Square Topology is Not Paracompact Space, $T'$ is not a paracompact space.

Hence the result.