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MATH 140A (Lecture B; course code 44775) Fall 2011 Instructor: Professor R. Reading Assignments (Note: All section or page numbers refer to the...

do problem III (a)

MATH 140A (Lecture B; course code 44775) Fall 2011 Instructor: Professor R. C. Reilly Assignment #6 Posted 11/03/2011 Remark This assignment covers the week of the midterm exam, October 31 through Friday November 5, so there is nothing much listed for the discussions. However, there is a ‘hand-in’ assignment, based on pre-midterm exam material, due on Wednesday November 9; see below. Reading Assignments (Note: All section or page numbers refer to the course text, “Elementary Analysis: The Theory of Calculus”, by K. Ross.) Monday 10/31/2011: Review for Midterm Exam on Wednesday 11/02/11 Wednesday 11/02/2011: MIDTERM EXAM Friday 11/04/2011: Section 11 Monday 11/07/2011: Section 11 Things to Prepare for the Discussions (These are not to be handed in.) Note: These are problems/examples/topics which the Teaching Assistant (TA) will cover in the discussion sections. You are expected to work on them before you go to the discussions. (A) (Items for the discussion on Tuesday 11/01/2011. Do not hand in.) REVIEW FOR MIDTERM EXAM (B) (Items for the discussion on Thursday 11/03/2011. Do not hand in.) GO OVER THE PROBLEMS ON THE MIDTERM EXAM Problems to be Handed in at Start of the Class on THURSDAY 11/10/2011 (Note the unusual hand-in day this time: Thursday, not Monday) (I) Let S be a bounded nonempty set of real numbers, and let T be the set of all numbers z of the form z = | x - y | , where x and y are in S . Problem (a) Show that the set T is also nonempty and bounded. (b) Show that sup T = (sup S ) - (inf S ). Remark This was the same as Problem # 6 on the ‘Practice Midterm Exam’. (II) (a) In Exercise 10.7 the text includes the hypothesis that sup S 6∈ S ; read the exercise for the details. Prove or Disprove The conclusion of Exercise 10.7 remains true if the hypothesis sup S 6∈ S is omitted from the statement. (b) Let S be a nonempty subset of IR such that S is not bounded above. Prove that there exists a nondecreasing sequence σ = ( s 1 ,s 2 , ...s n , ... ) of points in S such that lim n →∞ s n = sup S . More precisely, prove that σ can be chosen to be strictly increasing , in the sense that s n < s n +1 for each n . (Note that in Exercise 10.7 the text assumes that the sequence is bounded, so the result stated in this exercise is somewhat different.)
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(III) NOTE: This is a ‘Double-Value’ exercise, worth 8 points. Let σ = ( s 1 ,s 2 , ...s n , ... ) be a bounded sequence of real numbers. (a) Let L be a real number. Prove that the following statements are equivalent: (1) L = lim sup n →∞ s n . (2) For every ε > 0 there exists N 1 such that if n > N 1 , then s n < L + ε , but for every N 2 there exists at least one n > N 2 such that L - ε < s n . (b) Prove that lim inf n →∞ s n = - lim sup n →∞ ( - s n ). Remark Some texts use the condition described in Part (a)(2) as their definition of the limit superior, and use the formula in Part (b) as their definition of the limit inferior, at least in the case of bounded sequences. For such texts, the definitions of these concepts given in ‘Ross’ would then be theorems to be proved. (IV) Prove or Disprove If σ = ( s 1 ,s 2 , ...s n , ... ) is a sequence such that lim sup n →∞ s n = -∞ , then lim n →∞ s n exists.
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