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LECTURE 10: THE LEAST UPPER BOUND AXIOM
1. Introduction
Recall both R and Q are sets of numbers which obey the axioms P1 - P12; that
is, they are both examples of ordered fields.
So far most of our theory of limits and continuity of functions can equally be
applied in the context of functions on Q. We are interested in analysis over R,
however, and to proceed we will finally need introduce the additional property
which distinguishes R from Q.
This property, the least upper bound or completeness axiom, will ensure
that our real line R has?no “holes” and, in particular, we will deduce the existence
of expressions such as 2 as a consequence of it.
2. Suprema and infima
Definition. A set A
some x P R such that
„ R is bounded above (respectively, below) if there exists
a¤x
(respectively, a ¥ x)
for all a P A.
(1)
Any number x P R which satisfies (1) is called an upper (respectively, lower) bound
for A. If A is bounded both above and below, then A is said to be bounded.
Examples. Examples include:
The empty set is bounded.
Finite sets are bounded.
Intervals of finite length (e.g. pa, bq for a b) are bounded.
t1{n : n P Nu is bounded.
tlog n : n P Nu is bounded below; t log n : n P Nu is bounded above.
Intervals of the form p8, aq, p8, as are bounded above.
Intervals of the form pa, 8q, ra, 8q are bounded below.
Non-examples include:
N, Z, Q, R.
p8, as Y rb, 8q for any a ¤ b.
Notice that an upper bound is never unique.
Definition. A number x P R is the least upper bound or supremum of a set
A „ R if the following hold:
i) x is an upper bound of A;
ii) If y is an upper bound for A, the x ¤ y.
Definition. A number x P R is the greatest lower bound or infimum of a set
A „ R if the following hold:
i) x is an lower bound of A;
ii) If y is an lower bound for A, the x ¥ y.
Lemma (Uniqueness of suprema and infima). A set can have at most one supremum or infimum.
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LECTURE 10: THE LEAST UPPER BOUND AXIOM
Proof. Suppose x and y are suprema for A. Since y is an upper bound for A, if
follows that x ¤ y. Similarly, since x is an upper bound for A it follows that y ¤ x.
Hence x y, as required. Infima can be treated in the same manner.
Definition. If x is a supremum (respectively, infimum) of A, then we write sup A x (respectively, inf A x).
Examples.
1) For any a ¤ b it follows supra, bs b and inf ra, bs a.
2) For any a b it follows suppa, bq X Q b and inf pa, bq X Q a.
3) supt1{n : n P Nu 1 and inf t1{n : n P Nu 0.
4) Despite H being bounded, neither sup H nor inf H exist.
Recall, if there exists some a P A such that x ¤ a for all x P A, then we say x is
a maximum of A. It easy to see there can be at most one maximum of A and we
therefore write x max A.
Lemma. If A „ R and max A (respectively, min A) exists, then sup A
(respectively, inf A min A).
max A
3. The least upper bound axiom
P13: Least upper bound / completeness axiom. If A „ R is non-empty and
bounded above, then A has a supremum.
The properties P1 - P13 are referred to as the “axioms of a complete ordered
field”. For now, we will assume
R is a system of numbers which satisfies P1 - P13.
The idea is that these properties are “simple enough” that it appears natural to
study such an object. There are a number of objections to this, namely:
(1) We’ve just assumed R into existence. This seemed reasonable for N, Z, Q
which are definitely simpler objects to understand but perhaps in this case
we’ve stepped too far. It would be better if one could explicitly construct an
object from Q which satisfies P1 - P13. This would make us more confident
about the existence of R.
(2) There might be a number of many very different objects which satisfy P1
- P13; what right have we to talk about the real numbers?
It turns out that one can construct the reals (i.e. a system of numbers satisfying
P1 - P13) explicitly from the rationals; there are a number of ways to do this and
we will look at one in a later lecture. It also turns out that the reals are, in some
specific sense, the unique system of numbers which satisfy P1 - P13 but one has
to be careful how one defines “uniqueness” in this context. We won’t go into the
details of the uniqueness theorem in class, but if you’re interested you could read
Chapter 30 of Spivak.
The least upper bound axiom immediately distinguishes R from Q, viz.
Lemma. The least upper bound axiom fails over Q; that is, there exists some
A „ Q such that A is non-empty and bounded above but has no supremum in Q.
Proof. Simply let
A : tx P Q : x 0 or x2
2u.
Then 1 P A so that A H and 2 is an upper bound for A; indeed, if b ¥ 2, then
b2 ¥ 4 ¡ 2 so b R A. It remains to show A has no supremum in Q.
Suppose x P Q were a supremum for A and note 1 ¤ x, since 1 P A. Recall,
there does not exist any y P Q such that y 2 2 and so x2 2.
LECTURE 10: THE LEAST UPPER BOUND AXIOM
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If 1 ¤ x2
2, then there exists some n P N sufficiently large so that
2x
1
2 x2 .
n
n2
Now x 1{n P Q and px 1{nq2 x2 2x{n 1{n2 x2 2 x2 2 and,
consequently, x 1{n P A. But x x 1{n and so this contradicts the hypothesis
that x is an upper bound for A. From these observations one deduces that 2 x2
and so there exists some m P N sufficiently large so that
2x
x2 2.
m
Now x 1{m P Q and
px 1{mq2 px2 2q 2x{m 2 1{m2 ¡ 2 1{m2 ¡ 2.
Consequently, if a P A, then a x 1{m, so x 1{m is an upper bound for A.
But x 1{m x and so this contradicts the hypothesis that x is the least upper
bound for A.
Lemma. If A „ R is non-empty and bounded below, then A has an infimum.
4. The archimedean property revisited
In earlier lectures we assumed the following theorem.
Theorem (Archimedean property of N). The set N is unbounded.
Although intuitively obvious, strictly speaking we have not given a proof. It
turns out this requires the least upper bound axiom.
Corollary. For any ε ¡ 0 there exists some N
P N such that 1{n ε for all n ¥ N .
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]