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Introduction to Real Analysis Dr. Weihu Hong Clayton State University 10/2/2008 Cauchy Sequences { pn }n 1 in Definition 2.6.1 A sequence R is a Cauchy sequence if for every ε>0,there exists a positive integer no , such that pn pm for all integers n, m ≥ no . Remark. pn pm , n, m N and n, m no pn k pn , n N and n no , k N Theorem 2.6.2 (a) Every convergent sequence in R is a Cauchy sequence (b) Every Cauchy sequence is bounded. Theorem 2.6.3 If { pn }n 1 is a Cauchy sequence in R that has a convergent subsequence, then the sequence converges. { pn }n 1 Theorem 2.6.4 Every Cauchy sequence of real numbers converges. Remark. This theorem actually states that R is complete. Contractive sequences { p } n n 1 in R is Definition 2.6.6 A sequence contractive if there exists a real number b, with 0 < b < 1 such that | pn1 pn | b | pn pn1 | for all nєN with n≥2. Theorem 2.6.7 Every contractive sequence in R converges in R. Furthermore, if the sequence { pn }n 1 is contractive pn p, then and lim n b n 1 (a) | p pn | | p2 p1 |, and 1 b b (b) | p pn | | pn pn 1 | . 1 b where 0 b 1is the const in Definition 2.6.6.