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6.1
Definition (Absolute values.)
Recall that
if
is a real number, then the
absolute value of , denoted by
, is
defined by
We will assume the following properties of absolute value, that follow easily
from
the definition:
For all real numbers with
For all real numbers , and all
and

(6.2) 
We also have
and
6.3
Theorem.
Let
and let
. Then for all
we have
and
Equivalently, we can say that
and
Proof: I will prove only the first
statement. I have
6.4
Definition (Distance.)
The
distance between two
real numbers
and
is defined by
Theorem 6.3 says that the set of numbers whose distance
from is smaller than is the interval . Geometrically
this is clear from the picture.
I remember the theorem by keeping the picture
in mind.
Proof
For all and in R we have
and
so
Hence (Cf. (6.2))
6.7
Exercise.
Can you prove that for all
?
Can you prove that for all
?
Remark:
Let be real numbers with and .
Then
This result should be clear from the picture. We can give an analytic
proof as follows.
6.8
Examples.
Let
Then a number
is in
if and only if the distance from
to
is smaller than
, and
is in
if and only if the distance
from
to
is greater than
. I can see by inspection that
and
Let
If
, then
is in
if and only if
, i.e. if and only if
is closer to
than to
.
I can see by inspection that the point
equidistant from
and
is
, and that the numbers that are
closer to
than to
are the positive numbers, so
.
I can also do this analytically, (but in practice I wouldn't)
as follows. Since the alsolute values are all nonnegative
6.9
Exercise.
Express
each of the four sets below as an interval or a union
of intervals. (You can do this problem by inspection.)
6.10
Exercise.
Sketch the graphs of the functions from
to
defined by the
following equations:
(No explanations are expected for this problem.)
6.11
Exercise.
Let
be the functions described in the
previous exercise. By looking at the graphs, express each of the
following six sets in terms of intervals.
Let
. Represent
graphically on a number line.
Remark: The notation for absolute value of was introduced by
Weierstrass in 1841 [15][Vol 2,page 123]. It was first introduced
in
connection with complex numbers. It is surprising that analysis advanced so
far
without introducing a special notation for this very important function.
Next: 6.2 Approximation
Up: 6. Limits of Sequences
Previous: 6. Limits of Sequences
Index
Ray Mayer
20070907