Is there an extended definition of a Cauchy sequence? My prof wants one with a proof that a sequence divergent to infinity is Cauchy and vice versa. My first thought was that a sequence should be Cauchy if it is Cauchy in the real sense or else that for any M, there are nth and mth terms of the sequence such that that their difference is g.t. M and the difference between the nth and (m+j)th terms is g.t. M for j = 0,1,... . But this definition is dependent on whether the sequence approaches + or - infinity. Any ideas?