Cantor's diagonal argument

Cantor's diagonal argument proves that the infinite binary sequences form an uncountable set: a sequence obtained by changing the $n$th digit of the $n$th listed sequence differs from every sequence in the list.

Cantor's diagonal argument is a mathematical proof devised by Georg Cantor in the late 19th century. It demonstrates that not all infinities are equal, specifically showing that the set of real numbers is uncountably infinite and larger than the countably infinite set of natural numbers.