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Full Idea
Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers.
Gist of Idea
Irrational numbers are the limits of Cauchy sequences of rational numbers
Source
report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.92