This is a proof in terms of nets. Recall the following facts:

Lemma 1 - A net in converges to if and only if each coordinate converges to

Lemma 2 - A topological space is compact if and only if every net in has a convergent subnet.

Lemma 3 - Every net has a universal subnet.

Lemma 4 - A universal net in a compact space is convergent. (see this entry)

We now prove Tychonoff's theorem.

Proof (Tychonoff's theorem) : Let be a net in .

Using Lemma 3 we can find a universal subnet of .

It is easily seen that each coordinate net is a universal net in .

Using Lemma 4 we see that each coordinate net converges, because is compact.

Using Lemma 1 we see that the whole net converges in .

We conclude that every net in has a convergent subnet, so, by Lemma 2, must be compact.