noun convergent-sequence an infinite sequence, x 1 , x 2 , â€¦, whose terms are points in Ek, in which there exists a point y such that the limit as n goes to infinity of xn = y if and only if for every Îµ>0, there exists a number N such that i > N and j > N implies | xi âˆ’ xj |< Îµ. Also called Cauchy sequence, convergent sequence. Compare complete (def 10b). 1