Extended Real Number System: , and to extend the usual ordering on
by declaring that
for all
. The completeness of
can be stated as: Every subset of
of
has a least upper bound, or supremum, and a greatest lower bound, or infimum, which are denoted by
and
. If
, we also write:
,
.
From completeness it follows that every sequence in
has a limit superior and a limit inferior:
,
The sequence converges (in
) iff these two numbers are equal (and finite), in which case its limit is their common value. One can also define
and
for functions
, e.g.
.
The arithmetical operations on can be partially extended to
:
,
,
,
,
,
.
We employ the following notation for intervals in : If
,
,
,
,
.
If is an arbitrary set and
, we define
to be the supremum of its finite partial sums:
is finite $latex \}$.
Proposition. Given , let
. If
is uncountable, then
. If
is countably infinite, then
where
is any bijection and the sum on the right is an ordinary infinite series.
Proof. We have where
. If
is uncountable, then some
must be uncountable, and
for
a finite subset of
. If follows that
. If
is countably infinite,
is a bijection, and
, then every finite subset
of
is contained in some
. Therefore,
.
Taking the supremum over , then
,
and then taking the supremum over , we obtain the desired result.
Some terminology concerning (extended) real-valued functions: A relation between numbers that is applied to functions is understood to hold pointwise. Thus means that
for every
, and
is the function whose value at
is
. If
and
,
is called increasing if
whenever $x\le y$ and strictly increasing if
whenever
; similarly for decreasing. A function is either increasing or decreasing is called monotone.
If is an increasing function, then
has right- and left-hand limits at each point:
,
.
is called right continuous if
for all
and left continuous if
for all
.
For points or $\mathbb C$,
denotes the ordinary absolute value or modulus of
,
. For points
or
,
denotes the Euclidean norm:
.
We recall that a set is open if for every
,
includes an interval centered at
.
Proposition. Every open set in is countable disjoint union of open intervals.
Proof. If is open, for each
consider the collection
of all open intervals
such that
. It is easy to check that the union of any family of open intervals containing a point in common is again an open interval, and hence
is an open interval. It is the largest element of
. If $x, y\in U$ then either
or
, for otherwise
would be a larger open interval that
in
. Thus if
, the (distinct) members of
are disjoint, and
. For each
, pick a rational number
. The map
thus defined is injective, for if
then
; therefore
is countable.
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References:
[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 10-12.
[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.