"Definition: A sequence of real numbers (an) is Cauchy iff for all ε>0, there exists N s.t. n≥N and m≥N => |an-am|<ε. An equivalent definition is: for all ε>0, there exists N s.t. n≥N => |an-aN|<ε. " ============================================= I don't exactly see why these definitions are equivalent. One direction (from 1st one to 2nd one) is clear, we can just take m=N which is clearly ≥N. But how can we prove the converse (i.e. starting with the 2nd definition, prove the 1st)? Any help is much appreciated!