Finite Product of Sigma-Compact Spaces is Sigma-Compact

Theorem
Let $n \in \Z_+^*$ be a positive integer.

Let $\left \{{\left({X_i, \vartheta_i}\right): 1 \le i \le n}\right\}$ be a finite set of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod_{i=1}^n \left({X_i, \vartheta_i}\right)$ be the product space of $\left \{{\left({X_i, \vartheta_i}\right): 1 \le i \le n}\right\}$.

Let each of $\left({X_i, \vartheta_i}\right)$ be $\sigma$-compact.

Then $\left({X, \vartheta}\right)$ is also $\sigma$-compact.