Paracompactness is Preserved under Projections

Theorem
Let $I$ be an indexing set.

Let $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \left({S, \tau}\right) = \prod_{\alpha \mathop \in I} \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$.

Let $\operatorname{pr}_\alpha: \left({S, \tau}\right) \to \left({S_\alpha, \tau_\alpha}\right)$ be the projection on the $\alpha$ coordinate.

If $\left({S, \tau}\right)$ is paracompact, then each of $\left({S_\alpha, \tau_\alpha}\right)$ is also paracompact.