45 2.3 Monotone Sequences Definition 2.3.1 A sequence {an}is calledincreasingif an ≤an+1 for all n∈N. It is called decreasingif an ≥an+1 for all n∈N. If {an}is increasing or decreasing, then it is called a monotone sequence. The sequence is called strictly increasing (resp. strictly decreasing) if an <an+1 for all n∈N (resp. an >an+1 for all n∈N). If {an}is strictly increasing or strictly decreasing, then it is called a strictly monotone sequence. It is easy to show by induction that if {an} is an increasing sequence, then an ≤am whenever n≤m. Theorem 2.3.1 — Monotone Convergence Theorem. Let {an}be a sequence of real numbers. The following hold: (i) If {an}is increasing and bounded above, then it is convergent. (ii) If {an}is decreasing and bounded below, then it is convergent. Proof: (i) Let {an}be an increasing sequence that is bounded above. Define A={an : n∈N}. Then Ais a subset of Rthat is nonempty and bounded above and, hence, supAexists. Let ℓ =supAand let ε >0. By Proposition 1.5.1, there exists N∈Nsuch that ℓ−ε <aN ≤ℓ. Since {an}is increasing, ℓ−ε <aN ≤an for all n≥N. On the other hand, since ℓ is an upper bound for A, we have an ≤ℓ for all n. Thus, ℓ−ε <an ≤ℓ < ℓ+ε for all n≥N. Therefore, limn→∞an =ℓ. (ii) Let {an} be a decreasing sequence that is bounded below. Define bn =−an. Then {bn} is increasing and bounded above (if Mis a lower bound for {an}, then−Mis an upper bound for {bn}). Let ℓ =lim n→∞ bn =lim n→∞ (−an). Then {an}converges to −ℓ by Theorem 2.2.1. □ Remark 2.3.1 It follows from the proof of Theorem 2.3.1 that if {an} is increasing and bounded above, then lim n→∞ an =sup{an : n∈N}. Similarly, if {an}is decreasing and bounded below, then lim n→∞ an =inf{an : n∈N}.
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