Tychonoff's Theorem/General Case/Proof 1

Proof
First assume that $X$ is compact.

From Projection from Product Topology is Continuous, the projections $\pr_i : X \to X_i$ are continuous.

It follows from Continuous Image of Compact Space is Compact that the $X_i$ are compact.

Assume now that each $X_i$ is compact.

By Equivalent Definitions of Compactness it is enough to show that every ultrafilter on $X$ converges.

Thus let $\FF$ be an ultrafilter on $X$.

From Image of Ultrafilter is Ultrafilter, for each $i \in I$, the image filter $\map {\pr_i} \FF$ is an ultrafilter on $X_i$.

Each $X_i$ is compact by assumption.

So by Equivalent Definitions of Compactness, each $\map {\pr_i} \FF$ converges.

By Filter on Product Space Converges iff Projections Converge, $\FF$ converges.