Question: are all cauchy sequences monotone?
Answer: For example, one may consider S = [0,1) and T = (1,2]. ) is Cauchy. However, it is not either increasing or decreasing, since s1 > s2 and s2 < s3, so it is not monotone. ... We proved that every bounded sequence (sn) has a convergent subsequence (snk ), but all convergent sequences are Cauchy, so (snk ) is Cauchy.