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 x_{1}, a second number x_{2},
a third x_{3}, 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 x_{1}, x_{2}, x_{3}, 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 x_{i}
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 x_{n} = 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 x_{n}
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.