# Diagonal process

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A method of using a sequence consisting of sequences

$$\alpha _ {i} = ( a _ {i1} , a _ {i2} , . . . ) ,\ \ i = 1 , 2 \dots$$

to construct a sequence $\alpha = ( a _ {1} , a _ {2} , . . . )$ where $a _ {i} \neq a _ {ii}$ for any $i = 1 , 2 \dots$ or $a _ {i} = a _ {ii}$ for all $i$. 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 $[ 0, 1 ]$ 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 $E$, a sequence of functions converging on a countable subset of $E$.

The diagonal process of renumbering puts the multiple sequence $\{ a _ {ik} \}$, $i = 1 , 2 , . . .$; $k = 1 , 2 , . . .$ into correspondence with the sequence $a _ {11} , a _ {12} , a _ {21}$; $\dots$ $a _ {k1} , a _ {k-} 1,2 \dots a _ {k-} i,i+ 1 \dots a _ {1k}$; $\dots$ and is used, for example, in proving that the union of a countable set of countable sets is itself countable .

How to Cite This Entry:
Diagonal process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_process&oldid=50987
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article