# MATH 140A (Lecture B; course code 44775) Fall 2011

Question

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 ∈ S ; read the exercise for the

details.

Prove or Disprove The conclusion of Exercise 10.7 remains true if the hypothesis sup S ∈ S is

omitted from the statement.

(b) Let S be a nonempty subset of I such that S is not bounded above. Prove that there exists

R

a nondecreasing sequence σ = (s1 , s2 , . . . sn , . . . ) of points in S such that limn → ∞ sn = sup S .

More precisely, prove that σ can be chosen to be strictly increasing, in the sense that sn 0 there exists N1 such that if n > N1 , then sn N2 such that L − ε < sn .

(b) Prove that lim inf n → ∞ sn = −lim supn → ∞ (−sn ).

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 = (s1 , s2 , . . . sn , . . . ) is a sequence such that lim supn → ∞ sn = −∞,

then limn → ∞ sn exists.

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