MATH 4400
-
Introduction to Topology: Homework 4
1.
Prove the following properties of
ℤ
and
ℤ
+
:
a.
?, ? ∈ ℤ
+
⟹ ? + ? ∈ ℤ
+
.
[
Hint
: Show that given
? ∈ ℤ
+
, the set
𝑋 = {𝑥 | 𝑥 ∈ ℝ
and
? + 𝑥 ∈ ℤ
+
}
is inductive]
b.
?, ? ∈ ℤ
+
⟹ ? ∗ ? ∈ ℤ
+
.
2.
Show that every positive number
?
has exactly one positive square root, as follows:
a.
Show that if
𝑥 > 0
and
0 ≤ ℎ < 1
, then
(𝑥 + ℎ)
2
≤ 𝑥
2
+ ℎ(2𝑥 + 1),
(𝑥 − ℎ)
2
≥ 𝑥
2
− ℎ(2𝑥).
b.
Let
𝑥 > 0
. Show that if
𝑥
2
< ?,
then
(𝑥 + ℎ)
2
< ?
for some
ℎ > 0;
and if
𝑥
2
> ?,
then
(𝑥 − ℎ)
2
> ?
for some
ℎ > 0
.
c.
Given
? > 0
, let
?
be the set of all real numbers
𝑥
such that
𝑥
2
< ?.
Show that
?
is
bounded above and contains at least one positive number. Let
? = sup ?;
show that
?
2
= ?.
d.
Show that if
?
and
?
are positive and
?
2
= ?
2
,
then
? = ?
.
