Product of Countable Discrete Space with Sierpiński Space is Paracompact

Theorem
Let $T_X = \left({X, \vartheta}\right)$ be a countable discrete space.

Let $T_Y = \left({\left\{{0, 1}\right\}, \tau_0}\right)$ be a Sierpiński space.

Let $T_X \times T_Y$ be the product space of $T_X$ and $T_Y$.

Then $T_X \times T_Y$ is paracompact.