45 2.3 Monotone Sequences Defnition 2.3.1 A sequence {an} is called increasing if an ≤ an+1 for all n ∈ N. It is called decreasing if 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. Defne A = {an : n ∈ N}. Then A is a subset of R that is nonempty and bounded above and, hence, supA exists. Let ℓ = supA and let ε > 0. By Proposition 1.5.1, there exists N ∈ N such 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. Defne bn = −an. Then {bn} is increasing and bounded above (if M is a lower bound for {an}, then −M is an upper bound for {bn}). Let ℓ = lim bn = lim (−an). n→∞ n→∞ 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 an = sup{an : n ∈ N}. n→∞ Similarly, if {an} is decreasing and bounded below, then lim an = inf{an : n ∈ N}. n→∞
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