Tychonoff's Theorem

Let $X = \prod_{\alpha \in A} X_{\alpha}$ be a Tychonoff space. We will use the universal nets definition of compactness to prove $X$ is compact. Also, we use the fact that the net $\lbrace x_{\beta} \rbrace$ converges in $X$ iff each of its projections $\pi_{\alpha} (x_{\beta})$ converges.

If $f$ is a continuous mapping and $\lbrace x_{\beta} \rbrace$ is a universal net, then $f(x_{\beta})$ is universal. Therefore, because $\pi_{\alpha}$ is continuous for all $\alpha$, and beacuse $X$ is compact for all $\alpha$, we conclude that for all universal nets $\lbrace x_{\beta} \rbrace$, the projections $\pi_{\alpha}(x_{\beta})$ converge for all $\alpha$, and thus $\lbrace x_{\beta} \rbrace$ converges.