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Basic Analysis I — Real Analysis
· Problem 1.2.10
Real Numbers — Jiri Lebl
Mathematics — Real Analysis · supremum-infimum
Problem
Let
A
A
A
and
B
B
B
be two nonempty bounded sets of nonnegative real numbers. Define the set
C
:
=
{
a
b
:
a
∈
A
,
b
∈
B
}
C := \{ab : a \in A, b \in B\}
C
:=
{
ab
:
a
∈
A
,
b
∈
B
}
. Show that
C
C
C
is a bounded set and that
sup
C
=
(
sup
A
)
(
sup
B
)
and
inf
C
=
(
inf
A
)
(
inf
B
)
.
\sup C = (\sup A)(\sup B) \quad \text{and} \quad \inf C = (\inf A)(\inf B).
sup
C
=
(
sup
A
)
(
sup
B
)
and
in
f
C
=
(
in
f
A
)
(
in
f
B
)
.
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