is bounded above and below and has a minimum and a maximum,
(2)
S
is bounded above and below and has a minimum but not a maximum, (3)
S
is bounded above and below and has a maximum but not a minimum, (4)
S
is
bounded above and below and has neither a maximum nor a minimum, (5)
S
is
bounded below but not above and has a minimum, (6)
S
is bounded below but
not above and does not have a minimum, (7)
S
is bounded above but not below
and has a maximum, (8)
S
is bounded above but not below and does not have a
maximum, (9)
S
is not bounded above nor is bounded below.
Now show that each of these 9 cases leads to one of the 9 types described in
the third ‘bullet’ in Chapter 2.
I will do two as examples, the other seven are
similar. For example, in case (3), let
a
= inf(
S
) and
b
= sup(
S
) = max(
S
), then
S
⊂
(
a, b
]. Let
y
∈
(
a, b
]. By the ‘Principle’ on page 18, there exists an
x
∈
S
with
a
≤
x < y
≤
b
.
So
y
∈
S
since
x, b
∈
S
.
So every
y
∈
(
a, b
] is in
S
.
Therefore
(
a, b
]
⊂
S
, so
S
= (
a, b
], the third kind of interval.
In case (8), let
b
= sup(
S
), then
S
⊂
(
−∞
, b
). Let
y
∈
(
−∞
, b
). Since
S
is not
bounded below,
y
is not a lower bound for
S
, so there exists
x
∈
S
with
x < y
. By
the ‘Principle’ on page 18, there exists a
z
∈
S
with
y < z
≤
b
. So
y
∈
S
since
x, z
∈
S
. So every
y
∈
(
−∞
, b
) is in
S
. Therefore (
−∞
, b
)
⊂
S
, so
S
= (
−∞
, b
),
the eighth kind of interval.
square
Homework 2.
•
Text 5th Ed Section 3.2 number 3a,b,c, Section 3.3 numbers 1a-c, 3degi
(prove these), 4e,i (prove these), 5, 6, 7, 8 [4th (and 3rd) Ed numbers:
12.1a-c, 12.3d,e,g,i (prove these), 12.4e,i (prove these), 12.5, 12.6, 12.7,
12.8.]
22
MATH 3333–INTERMEDIATE ANALYSIS–BLECHER NOTES
•
(A) Show that every nonempty subset of
R
which is bounded below has
a greatest lower bound or inf (this falls out of the proof of Question 7(b)
above (or 12.7 (b)) with
k
=
−
1).
•
Also do case (4) and (9) in the proof of Theorem 3.13, in both directions
((
⇒
) and (
⇐
)).
3.4.
Open and closed sets.
In calculus you met some important definitions,
something like the following:
•
A
neighborhood
of
x
is an open interval centered at
x
. For example, (1
,
3)
is a neighborhood of 2.
Every neighborhood of
x
is of the form
N
(
x, ǫ
) = (
x
−
ǫ, x
+
ǫ
) =
{
y
∈
R
:
|
x
−
y
|
< ǫ
}
,
for some
ǫ >
0. [Picture drawn in class.]
We occasionally say that
ǫ
is the radius of
N
(
x, ǫ
).
•
A
deleted neighborhood
of
x
is a neighborhood of
x
with the midpoint
x
removed:
N
∗
(
x, ǫ
) = (
x
−
ǫ, x
)
∪
(
x, x
+
ǫ
) =
N
(
x, ǫ
)
\{
x
}
=
{
y
∈
R
: 0
<
|
x
−
y
|
< ǫ
}
.
[Picture drawn in class.]
•
An
interior point
of a set
S
is a point
x
∈
S
such that
S
contains some
neighborhood of
x
. That is,
∃
ǫ >
0 s.t.
N
(
x, ǫ
)
⊂
S
. Or equivalently,
∃
ǫ >
0
s.t. (
x
−
ǫ, x
+
ǫ
)
⊂
S
.
•
The
interior
of a nonempty set
S
is the set of all interior points of
S
. We
write this set as
S
◦
or Int(
S
).
This may be the empty set (if
S
has no
interior points).
For example if
A
= [1
,
3) then
A
◦
= (1
,
3).
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