therefore this limit is an element of the intersection that is above which shows that the intersection is unbounded. QED. From this, it can be seen that if is a regular cardinal, then is a non-principal -complete proper filter on the set (that is, on the poset ). If is a regular cardinal then club sets are also closed under diagonal intersection. In fact, if is regular and is any filter on closed under diagonal intersection, containing all sets of the form for then must include all club sets. See also References Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. . Lévy, Azriel (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. Ordinal numbers Set theory