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Introduction to Real Analysis Dr. Weihu Hong Clayton State University 9/18/2008 Theorem 2.3.2 {an }n 1 If is monotone and bounded, then it converges. { I } Corollary 2.3.3 If n n1 is a sequence of closed and bounded intervals with I n I n1 for all nєN, then I n n 1 Note: The intervals must be closed in Corollary 2.3.3 Infinite Limits {an }n 1 Definition 2.3.6 Let be a sequence of real numbers. We say that {an }n 1 approaches infinity, or that {an }n 1 diverges to ∞, denoted an or lim an n if for every positive real number M, there exists an integer KєN such that an M for all n K How would you define a sequence approaches to −∞? Theorem 2.3.7 {an }n 1 If is monotone increasing and not bounded above, then an as n . Proof: Since the sequence is not bounded above, therefore, for every positive number M, there exists a term aK such that aK M . Since the sequence is increasing, thus, an aK M for all n K Therefore, an as n . Subsequence of a sequence Definition 2.4.1 Given a sequence consider a sequence such that {nk } sequence { pn } . in R, of positive integers n1 n2 n3 sequence { p } nk { pn } . Then the is called a subsequence of the Examples of subsequences of a sequence Consider a sequence { pn } . Let {2k 1}k 1 ,{2k}k 1 be two sequences of positive integers. Then we have two subsequences { p2 k 1} and { p2k } of the sequence, one of which is consisting of all the terms from the sequence { pn } with odd indices while the other one is consisting of all the terms from the sequence { pn } with even indices. Subsequential limit of a sequence Given a sequence { pn } .Let a be either a real number or ±∞. We say that a is a subsequential limit of the sequence { pn } if there exists a subsequence { pnk } such that pnk a as k Example of subsequential limit Consider the sequence {1 (1) n }n 1 . Is a = 2 a subsequential limit of the sequence? Is a = 0 a subsequential limit of the sequence? n {( 1 ) n } Consider the sequence n 1 . Is a = +∞ a subsequential limit of the sequence? Is a = -∞ a subsequential limit of the sequence? Theorem 2.4.3 Let { pn } be a sequence in R. If { pn } converges to p, then every subsequence { pn } of k also converges to p. { pn }