Math H161 Ð Brown Ð Au04

Homework 1, due Wednesday, September 29, 2004

1. Analogous to our definitions of upper bound and least upper bound, formulate your own definitions of

(a)  lower bound of a nonempty set  S

(b)  greatest lower bound of a nonempty set   S

2.  Section 13.2 of our textbook discusses sequences of real numbers.  A sequence is simply a list of real numbers consisting of a first number  x1, a second number  x2, a third  x3, and so on, so that there is a real number associated with every positive integer 1, 2, 3, É   (the positive integers serves simply as the subscripts when the real numbers are written using the notation x1,  x2,  x3,  É .)   Real numbers may be duplicated at will in a sequence; you still have a sequence.

You can make the real numbers in a sequence into a set of real numbers by simply ignoring duplications and the order in which the numbers appear in the sequence.  For example, the sequence 1, 1, 1, É in which each xi equals 1  gives you the set  { 1 }, and the sequence 1, 0, 1, 0,  É1,  0, É  gives you the set  { 0. 1 }.  But the sequence 1, 1/2, 1/3, 1/4, É ,  in which each xn = 1/n,   gives you the set  {1, 1/2, 1/3, 1/4, É}  because there are no duplications at all.

(a)  Write the sets corresponding to the sequences (d) and (e) in Example 1 on page 432.

(b)  Use long division of polynomials as suggested in the last paragraph on p. 435 to obtain equation (2) on that page.

(c)  Use this formula  (2) [for the sum of a finite geometric progression] to express each xn in sequence (f) on page 432 as a single fraction.

(d) Continuing from part (c) above, write the set corresponding to sequence (f).

3. For each set you found in problem 2, (from sequences (d), (e), (f)), find the least upper bound of the set (if there is one) and the greatest lower bound of the set (if there is one).

4.  Same as problem 3, except use the sets corresponding to sequences (a), (b), and (c) on page 432.