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Lebl-Basic-Analysis-I
· Problem 1.3.7
Real Numbers
Mathematics · bounded-functions
Problem
Let
D
D
D
be a nonempty set. Suppose
f
:
D
→
R
f : D \to \mathbb{R}
f
:
D
→
R
and
g
:
D
→
R
g : D \to \mathbb{R}
g
:
D
→
R
are bounded functions. a) Show
sup
x
∈
D
(
f
(
x
)
+
g
(
x
)
)
≤
sup
x
∈
D
f
(
x
)
+
sup
x
∈
D
g
(
x
)
and
inf
x
∈
D
(
f
(
x
)
+
g
(
x
)
)
≥
inf
x
∈
D
f
(
x
)
+
inf
x
∈
D
g
(
x
)
.
\sup_{x \in D}\bigl(f(x) + g(x)\bigr) \leq \sup_{x \in D} f(x) + \sup_{x \in D} g(x) \qquad \text{and} \qquad \inf_{x \in D}\bigl(f(x) + g(x)\bigr) \geq \inf_{x \in D} f(x) + \inf_{x \in D} g(x).
x
∈
D
sup
(
f
(
x
)
+
g
(
x
)
)
≤
x
∈
D
sup
f
(
x
)
+
x
∈
D
sup
g
(
x
)
and
x
∈
D
in
f
(
f
(
x
)
+
g
(
x
)
)
≥
x
∈
D
in
f
f
(
x
)
+
x
∈
D
in
f
g
(
x
)
.
b) Find an example (or examples) where we obtain strict inequalities.
Related theory
Characteristic Functions And Moment Generating Functions
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