Theorem - Let be a non-empty set. Every net in has a universal subnet. That is, there is a subnet such that for every either the subnet is eventually in or eventually in .

Proof : Let be a section filter for the net .

Let . is a directed set under the order relation given by

The map defined by is order preserving and cofinal. Therefore there is a subnet of associated with the map (that is, ).

We now prove that is a universal net.

Let . We have that is frequently in or frequently in .

Suppose is frequently in .

Let and . We have that by definition of section filter.

As is a filter, and so there exists with such that . Hence, .

As is frequently in , there exists with such that .

Also, is in , and therefore, in . So .

We conclude that for every . Therefore, generates a filter in . As is a maximal filter we conclude that , and consequently, .

We can now see that for every with , is in and so is in . Therefore, is eventually in .

Remark: If is frequently in , by an analogous argument we can conclude that it is eventually in .

This proves that is a universal subnet of .