Introduction to Mathematical Analysis I - Second Edition

41 (a) If { a n } and { b n } are convergent sequences, then { a n + b n } is a convergent sequence. (b) If { a n } and { b n } are divergent sequences, then { a n + b n } is divergent sequence. (c) If { a n } and { b n } are convergent sequences, then { a n b n } is a convergent sequence. (d) If { a n } and { b n } are divergent sequences, then { a n b n } is a divergent sequence. (e) If { a n } and { a n + b n } are convergent sequences, then { b n } is a convergent sequence. (f) If { a n } and { a n + b n } are divergent sequences, then { b n } is a divergent sequence. 2.3 MONOTONE SEQUENCES Definition 2.3.1 A sequence { a n } is called increasing if a n ≤ a n + 1 for all n ∈ N . It is called decreasing if a n ≥ a n + 1 for all n ∈ N . If { a n } is increasing or decreasing, then it is called a monotone sequence. The sequence is called strictly increasing (resp. strictly decreasing) if a n < a n + 1 for all n ∈ N (resp. a n > a n + 1 for all n ∈ N ). It is easy to show by induction that if { a n } is an increasing sequence, then a n ≤ a m whenever n ≤ m . Theorem 2.3.1 — Monotone Convergence Theorem. Let { a n } be a sequence of real numbers. The following hold: (a) If { a n } is increasing and bounded above, then it is convergent. (b) If { a n } is decreasing and bounded below, then it is convergent. Proof: (a) Let { a n } be an increasing sequence that is bounded above. Define A = { a n : n ∈ N } . Then A is a subset of R that is nonempty and bounded above and, hence, sup A exists. Let ` = sup A and let ε > 0. By Proposition 1.5.1 , there exists N ∈ N such that ` − ε < a N ≤ `. Since { a n } is increasing, ` − ε < a N ≤ a n for all n ≥ N . On the other hand, since ` is an upper bound for A , we have a n ≤ ` for all n . Thus, ` − ε < a n < ` + ε for all n ≥ N . Therefore, lim n → ∞ a n = ` . (b) Let { a n } be a decreasing sequence that is bounded below. Define b n = − a n .