Homework #5
Ordered Fields and the Real Numbers
Read section 1.7 and 1.8
Exercises from the text: Pages 47 & 48: #7, #8, #9, #13, #15
Pages 52 & 53: 11, 13,20
Pages 55-57: look at all exercises
Other exercises in thought and logic that you will most certainly find interesting.
1. Let A and B be non-empty, bounded subsets of
that sup( A B)  Max{sup( A), sup( B)}
. Show that A B is bounded and
2. Prove that no order can be defined on the set of complex numbers that makes the set
an ordered field (using the ordinary addition and multiplication in ).
3. If A, B  are non-empty, bounded sets, define A  B  {a  b : a  A and b  B} .
Prove that sup( A  B)  sup ( A)  sup ( B ) . Can equality ever hold? Can inequality
ever hold?
4. Prove that the rational numbers are dense in the reals. That is, between any two
distinct real numbers there is a rational number. (In fact, there are infinitely many).
5. Prove that if   S  is bounded, there is a smallest closed interval I that contains
S. That is, if J is any closed interval containing S, then J contains I.
6. Is #5 true if “closed interval” is replaced with “open interval”?