5.2.8 Guided Discovery: The Monotone Subsequence Theorem 1 Guided Discovery: The Monotone Subsequence Theorem 1. Making a conjecture a) Of the sequences below, some are bounded and some are unbounded, some converge and some diverge, but none of them is monotone. • Which of them has a monotone subsequence? K (2, , 3, , 4, , …) J L (3, 2, 1, 6, 5, 4, 9, 8, 7, ...) T ( …) Solution: Every sequence listed above has a monotone subsequence. In fact, each of them has infinitely many monotone subsequences! b) Find two sequences that do not have a monotone subsequence, one bounded and one unbounded. • Does your bounded sequence converge? • Does your unbounded sequence tend to infinity? Solution: This may take more than a few minutes, since it is impossible. Every sequence, bounded or unbounded, convergent or divergent, or whatever it may be, has a monotone subsequence: Monotone Subsequence Theorem: Every sequence has a monotone subsequence. Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 5.2.8 Guided Discovery: The Monotone Subsequence Theorem 2 2. Developing a proof: Looking for peaks Let be a sequence of real numbers. A term will be called a peak of the sequence when it is never exceeded by any term that follows ( ). In the following diagram, the tips of the “mountains” represent the terms of the sequence. The first five peaks are marked, where the arrows indicate the “line of sight” from each peak over the mountain range. Since is a peak, no mountain tip after is higher than . a) Mark the peaks in the diagrams below, where the tips of the “mountains” represent the terms of the sequence. Constant sequence: Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 5.2.8 Guided Discovery: The Monotone Subsequence Theorem 3 Strictly increasing sequence: Strictly decreasing sequence: Another sequence (where no term is larger than the tallest tip shown) Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 5.2.8 Guided Discovery: The Monotone Subsequence Theorem 4 b) Find an example of each of the following! • A sequence that has infinitely many peaks • A sequence that does not have any peaks • A sequence that has exactly three peaks Solution: Any decreasing sequence has infinitely many peaks (this includes constant sequences). Any strictly increasing sequence has no peaks at all. Here is an example of a sequence with exactly three peaks: Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 5.2.8 Guided Discovery: The Monotone Subsequence Theorem 5 3. Proof of the Monotone Subsequence Theorem Let be a sequence of real numbers. We will show that has a subsequence that is monotone. Case 1: Suppose has infinitely many peaks, xp1 ≥ xp2 ≥ xp3 ≥ xp4 Then the subsequence is ____________ and therefore monotone. Case 2: Suppose has only finitely many peaks (which may be none at all). Let be the final peak in the sequence. Then for all , _________ is not a peak. We now construct a monotone subsequence Let Then . is not a peak since ______________________________________ . So there exists Since . such that is not a peak, there exists ______ . _____ such that ______ . Continuing in this way, we obtain a _______________________________ (and therefore ______________________ ) subsequence . • The argument above can be made rigorous by replacing “continuing in this way” with a formal proof by ______________________________ . This completes the proof of the _____________ ____________ Theorem. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810
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