MthT 430 Term Project 2007

This project should be completed by a group of not less than two nor more than four persons. The group should turn in one typeup of the project. The paper should include a description of each member's contributions to the project.

The project should be typed. For suggestions on typing, see

http://www2.math.uic.edu/~jlewis/mtht430/430type.pdf

Assignment due dates:

November 21, 2007 - 6 PM: Progress Report - A note to jlewis@uic.edu on your progress on the project - include the names of the members of your group.

November 28, 2007 - 5 PM: Completed typed project due.

I. Warmup - Inequalities Again and a Useful Fact

1.

Show that if x and y are numbers, then x ≤ y if and only if for every ϵ > 0, x < y + ϵ.

2.

Let A be a bounded set of numbers. Define

−A = {−x | x ∈ A }.

Show that

inf A = − sup (−A).

II. Understanding sup and inf - Equivalent Definitions

It is useful to note a working characterization of supA (A ≠ ∅, A bounded):

If A ≠ ∅, supA is a number α such that ⎧ ⎪ ⎨ ⎪ ⎩ For every x ∈ A, x ≤ α, and For every ϵ > 0, there is an x ∈ A such that x > α− ϵ.

The first condition means that α is an upper bound for A . The second condition means for every ϵ > 0, α− ϵ is not an upper bound for A .

3.

Let A be a nonempty set of numbers which is bounded above. Show that

b = sup A

if and only if for every ϵ > 0

⎧ ⎪ ⎨ ⎪ ⎩ x < b + ϵ for all x ∈ A, and x > b − ϵ for some x ∈ A.

4.

Let A be a nonempty set of numbers which is bounded below. Show that

b = inf A

if and only if for every ϵ > 0

⎧ ⎪ ⎨ ⎪ ⎩ x > b − ϵ for all x ∈ A, and x < b + ϵ for some x ∈ A.

III. Adding sup and inf

5.

(See Chapter 8 - Problem 13) Let A and B be two nonempty sets of numbers which are bounded (both above and below). Define

A+ B = {x | x = a + b, a ∈ A, b ∈ B }.

Show that

sup (A + B) = sup A + sup B.

Show that

inf (A + B) = inf A + inf B.

IV. More Adding sup and inf

(See also Spivak, Chapter 8, Problem 9.) If f is a bounded function on [0,1], we define

sup f = sup x ∈ [0,1]  f(x) ≡ sup {f(x) | x ∈ [0,1]} inf f = inf x ∈ [0,1]  f(x) ≡ inf {f(x) | x ∈ [0,1]}

6.

(Easy - but not the same as Spivak Chapter 8 - Problem 13.) Show that if f and g are bounded functions on [0,1], then

sup (f + g) ≤ sup f + sup g.

7.

Give an example of a pair f and g of bounded functions on [0,1] such that

sup (f + g) < sup f + sup g.

8.

(Easy) Show that if f and g are bounded functions on [0,1], then

inf f + inf g ≤ inf (f + g).

9.

Show that if f and g are bounded functions on [0,1], then

inf f + sup g ≤ sup (f + g).

10.

Give an example of a pair f and g of bounded functions on [0,1] such that

inf f + sup g < sup (f + g).

11.

(Easy - See the previous problem(s).) Show that a pair f and g are bounded functions on [0,1], then

inf (f + g) ≤ inf f + sup g .

12.

(Monster Counterexample to Equality) Your group has shown: For a pair f and g of bounded functions on [0,1],

inf f + inf g ≤ inf (f + g) ≤ inf f + sup g ≤ sup (f + g) ≤ sup f + sup g.

Give an example of a pair f and g of bounded functions on [0,1] such that

inf f + inf g < inf (f + g) < inf f + sup g < sup (f + g) < sup f + sup g.

N.B. The string of inequalities is related to similar inequalities regarding limsup and liminf in

http://www2.math.uic.edu/~jlewis/mtht430/chap8fproj.pdf

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