Full Idea
The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient.
Gist of Idea
Cauchy gave a necessary condition for the convergence of a sequence
Source
Shaughan Lavine (Understanding the Infinite [1994], 2.5)
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.33