Consider the two-element set $\{0,\ 1\}$ equipped with the discrete topology, and form the countably infinite product

$\displaystyle X:=\{0,\ 1\}^\omega=\prod\limits_{n\in\mathbb{Z_+}}\{0,\ 1\}$

So $X$ consists of the infinite sequences $\displaystyle(x_n)_{n\in\mathbb{Z_+}}$, where for each $k\in\mathbb{Z}_+$, the $k$th term $x_k$ is either $0$ or $1$. Equip $X$ with the product topology.

Show that $X$ is compact (you may not use Tychonoff’s theorem).