MthT 430 Projects Chapter 8d Adding sup and inf
MthT 430 Projects Chapter 8d Adding sup and inf
It is useful to note a working characterization of supA (A
¹
Æ
):
If A
¹
Æ
, supA is the number
a
such that
ì
ï
í
ï
î
For
every
x
Î
A
,
x
£
a
,
and
For
every
e
>
0
,
there
is
an
x
Î
A
such
that
x
>
a
-
e
.
The first condition means that
a
is an upper bound for A
. The second condition means
for every
e
> 0,
a
-
e
is not an upper bound for A
.
1.
Show that if x and y are numbers, then x
£
y if and only if for every
e
> 0, x < y +
e
.
2.
Let A be a nonempty set of numbers which is bounded above. Show that
b =
sup
A
if and only if for every
e
> 0
ì
ï
í
ï
î
x
<
b
+
e
for
all
x
Î
A
,
and
x
>
b
-
e
for
some
x
Î
A
.
3.
Let A be a nonempty set of numbers which is bounded below. Show that
b =
sup
A
if and only if for every
e
> 0
ì
ï
í
ï
î
x
>
b
-
e
for
all
x
Î
A
,
and
x
<
b
+
e
for
some
x
Î
A
.
4.
(See Chapter 8 - Problem 13) Let A and B be two nonempty sets of numbers which are bounded (both above and below). Define
A+ B = {x | x = a + b, a
Î
A, b
Î
B }.
Show that
sup
(A + B) =
sup
A +
sup
B.
Show that
inf
(A + B) =
inf
A +
inf
B.
File translated from T
E
X by
T
T
H
, version 3.78.
On 29 Oct 2007, 19:27.