Supremum and Infimum
UBC M220 Lecture Notes by Philip D. Loewen
The Real Number System.
Work hard to construct from the axioms a set
R
with special elements
O
and
I
, and a subset
P
⊆
R
, and mappings
A
:
R
×
R
→
R
,
M
:
R
×
R
→
R
, for which defining the basic operations above in terms of
x
+
y
=
A
(
x, y
)
,
x
·
y
=
M
(
x, y
)
,
x >
O
⇔
x
∈
P
produces a consistent setup in which the familiar rules of arithmetic all work.
Trichotomy.
For every real number
x
, exactly one of the following is true:
x <
0
,
x
= 0
,
x >
0
.
By taking
x
=
b

a
, we deduce that whenever
a, b
∈
R
, exactly one of the following
is true:
a < b,
a
=
b,
a > b.
Given
a, b
∈
R
, it’s rather obvious that
a > b
=
⇒ ∃
ε >
0 :
a
≥
b
+
ε.
(Indeed, if
a > b
then
ε
=
a

b
obeys the conclusion.) The contrapositive of this
statement is logically equivalent, but occasionally useful:
∀
ε >
0
, a < b
+
ε
=
⇒
a
≤
b.
It reveals that one way to prove “
a
≤
b
” is to prove a collection of apparently easier
inequalities involving a larger righthand side.
Definition.
Let
S
⊆
R
. To say, “
S
is bounded above,” means there exists
b
∈
R
such that
(
*
)
∀
s
∈
S, s
≤
b.
Any number
b
satisfying (
*
) is called “an upper bound for
S
.”
Changing “
≤
” to “
≥
” in (
*
) produces a definition for the phrases “
S
is bounded
below” and “
b
is a lower bound for
S
”.
To say that
S
is bounded means that
S
is bounded above and
S
is bounded below.
Definition.
Let
S
⊆
R
. The phrase, “
β
is a least upper bound for
S
,” means two
things:
(i)
∀
s
∈
S, s
≤
β
, i.e.,
β
is an upper bound for
S
, i.e.,
(
*
)
∀
s
∈
S, s
≤
β.
File “sup2009”, version of 5 June 2009, page 1.
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2
P
HILIP
D. L
OEWEN
(ii) Every real number less than
β
is
not
an upper bound for
S
. Express this second
condition as
(
**
)
∀
ε >
0
,
∃
s
∈
S
:
β

ε < s.
Notation.
A given set
S
can have at most one least upper bound (LUB).
[Pf:
Suppose
β
0
is a least upper bound for
S
.
Any real
β > β
0
breaks (
**
)—
use
ε
=
β

β
0
and recall (
*
).
Any real
β < β
0
breaks (
*
)—use
ε
=
β
0

β
and
recall (
**
).]
If
S
has a least upper bound, it is a unique element of
R
denoted
sup
S
(Latin
supremum
).
A symmetric development leads to the concepts of sets bounded below, greatest
lower bounds, and the Latin notation
inf
S
(Latin
infimum
).
The Least Upper Bound Property.
The hard work in the axiomatic construction
of the real number system is in arranging the following fundamental property:
For each nonempty subset
S
of
R
, if
S
has an upper bound, then
there exists a unique real number
β
such that
β
= sup
S
.
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 Spring '10
 Loewen
 Supremum, Order theory, upper bound, PHILIP D. LOEWEN, LUB Property

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