This preview shows pages 1–3. Sign up to view the full content.
Chapter 1 – Calculus Review
Pg 3
Of greatest importance to us is that the set R of real numbers is
complete
 in more than one
sense! First, recall that a subset A of IR is said to be
bounded above
if there is some x
∈
IR such
that a
≤
x for all a
∈
A. Any such number x is called an
upper bound
for A.
Pg 3
The Least Upper Bound Axiom (sometimes called the completeness axiom).
Any nonempty set of real numbers with an upper bound has a least upper bound.
Pg 4
Least Upper Bound of a set A (notation)
Pg 4 – Example 1.1
sup(
∞
, 1) = 1 and sup{2  (1/n) : n = 1,2, .
..} = 2. Notice, please, that sup A is not necessarily an
element of A.
Pg 4
An immediate consequence of the least upper bound axiom is that we also have
greatest lower
bounds (g.l.b.)
, just by turning things around. The details are left as Exercise 1.
Pg 4 – Exercise 1
If A is a nonempty subset of JR that is bounded below, show that A has a greatest
lower bound.
That is, show that there is a number m
∈
IR satisfying: (i) m is a lower bound for A; and (ii) if x
is a lower bound for A, then x < m. [Hint: Consider the set A = {a: a E A} and show that
m = sup( A) works.]
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPf:
Notice that the greatest lower bound has the “opposite” definition of the least upper
bound, so we can consider the set obtained by “reflecting” the set A about the number 0. So, I
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 Higdon
 Calculus, Real Numbers

Click to edit the document details