Math 409-502Harold P. [email protected] Math Club MeetingMonday, September 207:00pm in Blocker 627Speakers:Dr. Philip Yasskin, “Rascal’s Triangle”Ms. Edith Andrews, Jane Long Middle School HOSTS programFREE FOODMath 409-502 September 20, 2004 — slide #2More about subsequences and convergenceMain theorems in Chapter 6• Nested interval theorem• Bolzano-Weierstrass theoremMain concepts in Chapter 6• cluster point• Cauchy sequence• supremum• lim supMath 409-502 September 20, 2004 — slide #3Nested intervalsTheorem. If the closed intervals[a1, b1] ⊇ [a2, b2] ⊇ · · · ⊇ [an, bn] ⊇ . . .are nested, then the intersection∞\n= 1[an, bn] is not empty.Moreover, if length[an, bn] → 0, then there is exactly one point common to all the intervals.Examples• The nested intervals [−1 − 1/n, 1 + 1/n] have intersection [−1, 1].• The nested intervals [1 − 1/n, 1] have intersection {1}.• The nested open intervals (0, 1/n) have empty intersection.Math 409-502 September 20, 2004 — slide #4Bolzano-Weierstrass theoremTheorem. A bounded sequence of real numbers has convergent subsequences.Proof: repeated bisection and the nested interval theorem.Examples• The sequence {sin n}∞n= 1has convergent subsequences.• Let xnbe the right-most digit of the nth prime number. Then the sequence {xn}∞n= 1hasconvergent subsequences.Math 409-502 September 20, 2004 — slide #5Cluster pointsDefinitionA cluster point of a sequence is the limit of a convergent subsequence. (Another name for thesame concept is accumulation point.)Examples• The sequence {(−1)n}∞n= 1has two cluster points: namely 1 and −1.• The sequence {n sin(nπ/2)}∞n= 1has one cluster point: namely 0.Math 409-502 September 20, 2004 — slide #6Homework• Read sections 6.1–6.3, pages 78–83.• Do Exercises 6.2/1 and 6.3/1 on pages 89–90.Math 409-502 September 20, 2004 — slide
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