Chapter 2
Construction of the real numbers
The completeness of
R
.
1
We have seen that
Q
and
R
are both ordered fields, so what is the difference? Topology:
R
is connected,
Q
is not. For example,
√
2 is not rational:
Q
has a “hole” at
√
2.
In topology, a set
X
is defined to be
connected
iff there are two nonempty open sets
A, B
such that
A
∪
B
=
X
and
A
∩
B
=
∅
. Example:
Q
= (
∞
,
√
2)
∪
(
√
2
,
∞
)
.
R
cannot be written in such a way.
Another way to phrase this:
completeness
. Let
A
⊆
(
b, c
). Then
∃
x
∈
R
such that
1.
x
is an upper bound for
A
:
∀
a
∈
A, a
≤
x
.
2.
y
is an upper bound for
A
=
⇒
x
≤
y
.
We say
x
is the
least upper bound
for
A
or
supremum
of
A
, and write
x
= sup
A
.
Example: there is no “smallest rational number” that is larger than (or at least as
large as) every element of
A
= (0
,
√
2).
1
May 2, 2007
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30
Math 413
Construction of the real numbers
2.1
Cauchy sequences
Definition 2.1.1.
Let
{
x
n
}
be a sequence in
Q
. We say the
limit
of
{
x
n
}
is
L
(or that
{
x
n
}
converges
to
L
) iff
For each
m
= 1
,
2
, . . . ,
∃
N
m
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 Fall '11
 Wong
 Topology, Real Numbers, Cauchy sequence, Xn, Nonexample

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