Paracompactness is Preserved under Projections

Theorem
Let $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$ be a sequence of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$.

Let $\operatorname{pr}_\alpha: \left({X, \vartheta}\right) \to \left({X_\alpha, \vartheta_\alpha}\right)$ be the projection on the $\alpha$ coordinate.

Then if $\left({X, \vartheta}\right)$ is paracompact, then each of $\left({X_\alpha, \vartheta_\alpha}\right)$ is also paracompact.