Full Idea
Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers.
Gist of Idea
A real is associated with an infinite set of infinite Cauchy sequences of rationals
Source
report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.38
A Reaction
This sounds remarkably like the endless decimals we use when we try to write down an actual real number.