Theorem - A universal net in a compact space is convergent.

Proof : Suppose by contradiction that was not convergent. Then for every we would find neighborhoods such that

The collection of all this neighborhoods cover , and as is compact, a finite number also cover .

The net is not eventually in so it must be eventually in (because it is a universal net). Therefore we can find such that

Because we have a finite number we can find such that for each .

Then for all , i.e. for all . But cover and thus we have a contradiction.