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Lebl-Basic-Analysis-I
· Problem 1.3.5
Real Numbers
Mathematics · bounded-functions
Problem
Let
f
:
D
→
R
f : D \to \mathbb{R}
f
:
D
→
R
and
g
:
D
→
R
g : D \to \mathbb{R}
g
:
D
→
R
be functions (
D
D
D
nonempty). a) Suppose
f
(
x
)
≤
g
(
y
)
f(x) \leq g(y)
f
(
x
)
≤
g
(
y
)
for all
x
∈
D
x \in D
x
∈
D
and
y
∈
D
y \in D
y
∈
D
. Show that
sup
x
∈
D
f
(
x
)
≤
inf
x
∈
D
g
(
x
)
.
\sup_{x \in D} f(x) \leq \inf_{x \in D} g(x).
x
∈
D
sup
f
(
x
)
≤
x
∈
D
in
f
g
(
x
)
.
b) Find a specific
D
D
D
,
f
f
f
, and
g
g
g
, such that
f
(
x
)
≤
g
(
x
)
f(x) \leq g(x)
f
(
x
)
≤
g
(
x
)
for all
x
∈
D
x \in D
x
∈
D
, but
sup
x
∈
D
f
(
x
)
>
inf
x
∈
D
g
(
x
)
.
\sup_{x \in D} f(x) > \inf_{x \in D} g(x).
x
∈
D
sup
f
(
x
)
>
x
∈
D
in
f
g
(
x
)
.
Related theory
Characteristic Functions And Moment Generating Functions
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