Problem Statement:

Suppose that { } is a Cauchy sequence and that there is a subsquence { } and a number such that . Show that the full sequence converges, too; that is .

My Solution:

It will suffice to prove that { } converges to if and only if every subsequence of { } converges to .

( )Suppose . Consider { } { }.

Let . For some , . Then if . so .

( ){ } is a subset of itself, so .