This proof can be considered a variation of the [proof using ultrafilters on $X$](https://mathoverflow.net/questions/26416/what-is-your-favorite-proof-of-tychonoffs-theorem/26430#26430). I want mainly to point out that we can avoid transferring the ultrafilters through the projections if we use a slightly more general characterization of compactness using ultrafilters.$\newcommand{\FF}{\mathcal F}\newcommand{\UU}{\mathcal U}$

**Definition.** Let $X$ be a topological space, $x\in X$, $f\colon M\to X$ be a function and $\FF$ be a filter on $X$. Then we say that $x$ is _an $\FF$-limit of $f$_ iff for every neighborhood $U\ni x$ we have $$f^{-1}[U]\in\FF.$$

Basically, this definition says that $f^{-1}[U]$ has to be a "big set". (You can compare this with the definition of a limit of a sequence $f\colon\mathbb N\to X$ where $f^{-1}[U]$ has to be a cofinite set, i.e., it belongs to the Fréchet filter.)

Some references concerning this notion can be found in this post: [Where has this common generalization of nets and filters been written down?](https://math.stackexchange.com/q/1568548) 

We can now characterize compactness in the following way

**Fact.** A topological space is compact if and only if for every function $f\colon M\to X$ and every ultrafilter $\UU$ on $M$ there exists an $\UU$-limit in $X$.

A proof of the "easy" implication can be found, for example, here: [Basic facts about ultrafilters and convergence of a sequence along an ultrafilter](https://math.stackexchange.com/q/51476). Proof of the fact that this characterizes compact spaces will probably depend on the facts which were already proven about compact spaces and (ultra)filters at this point.

**Proof of Tychonoff theorem.** Let $X=\prod\limits_{i\in I} X_i$ be a product of compact spaces. Suppose we have an ultrafilter $\UU$ on $M$ and a function $f\colon M\to X$. Then for each $i\in I$ there exists som $\UU$-limit of $p_i\circ f$ in the compact space $X_i$. Then the point $x$ determined by $p_i(x)=x_i$ is an $\UU$-limit of $f$ in $X$.

(In the proof, we have also used the fact $\FF$-limit in topological product corresponds to pointwise $\FF$-limits for each $i\in I$.) 

Proof of Tychonoff's theorem along these lines is given, for example, in Dixmier's _General Topology ([zbmath 0545.54001](https://zbmath.org/?q=an:0545.54001), [MR753644](https://mathscinet.ams.org/mathscinet-getitem?mr=753644)) as [Theorem 4.3.6](https://books.google.com/books?id=VQHqBwAAQBAJ&pg=PA44).