# Diagonal process

A method of using a sequence consisting of sequences

to construct a sequence where for any or for all . The diagonal process was first used in its original form by G. Cantor

in his proof that the set of real numbers in the segment is not countable; the process is therefore also known as Cantor's diagonal process. A second form of the process is utilized in the theory of functions of a real or a complex variable in order to isolate, out of a family of bounded functions on a set , a sequence of functions converging on a countable subset of .

The diagonal process of renumbering puts the multiple sequence , ; into correspondence with the sequence ; ; and is used, for example, in proving that the union of a countable set of countable sets is itself countable [2].

#### References

 [1a] G. Cantor, "Ueber eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen" , Gesammelte Abhandlungen , G. Olms (1932) pp. 115–118 [1b] G. Cantor, "Ueber eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen" J. Reine Angew. Math. , 77 (1874) pp. 258–262 [2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [3] R. Péter, "Rekursive Funktionen" , Ungar. Akad. Wissenschaft. (1957) [4] S.C. Kleene, "Mathematical logic" , Wiley (1967) [5] J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)