Product of Countable Discrete Space with Sierpiński Space is Paracompact

Theorem
Let $T_X = \struct {S, \tau}$ be a countable discrete space.

Let $T_Y = \struct {\set {0, 1}, \tau_0}$ 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.

Proof
From Discrete Space is Paracompact, $T_X$ is paracompact.

We have that the Sierpiński space $T_Y$ is a finite topological space.

From Finite Topological Space is Compact, $T_Y$ is a compact space.