Paracompactness is Preserved under Projections

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 $\pr_\alpha: \struct{S, \tau} \to \struct{S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate.

If $\struct{S, \tau}$ is paracompact, then each of $\struct{S_\alpha, \tau_\alpha}$ is also paracompact.