Math H161 Ð Brown Ð Au03

Homework 1, due October 1, 2003

1. Formulate your own definitions, analogous to our definitions of upper bound and least upper bound, of

(a) **lower bound** of a nonempty set S

(b) **greatest lower bound**
of a nonempty set S

Hint for (a): try to come up with a suitable definition for sets on your own before you look in the index of our textbook to find its definition for sequences.

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 integer serves as
the subscript when the real numbers are written using the notation x_{1}, x_{2}, x_{3}, É Real numbers may be repeated at will in a
sequence, and you still have a sequence.
On the other hand, 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 gives you the set { 1 }, and the sequence 1, 0, 1,
0, É 0, É gives you the set { 0. 1 }. But the sequence x_{n},
= 1/n, i.e., 1, 1/2, 1/3, 1/4, É gives you the set {1, 1/2, 1/3, 1/4, É] because there are no duplications at
all.

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

(b) Use the
formula for the sum of a geometric series to express each x_{n}, in sequence (f) in a shorter form.

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

3. For the sets you found in 2. above, for each of the sets for 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).

OVER for problem 4.

4. Let f(x) = x^{3}.

(a) Sketch a graph of this function from x = 0 to x = 1,

(b) Select
three points x_{1}, x_{2}, x_{3}, which will break up the closed interval
[0, 1] into four subintervals for
which the length of the longest subinterval is less that 0.3.

(c) Using your subintervals, find the lower sum and the upper
sum which approximate the definite
integral from 0 to 1 of x^{3}.

(d) Same as (b) and (c) except select five intermediate points instead of three. (Still keep the length of the longest subinterval less that 0.3.) Look at the graph of the function and choose the points in to get better approximations for your lower and upper sums then equally spaced intermediate points would give.