MthT 430 Projects Chapter 1a Inequalities
MthT 430 Projects Chapter 1a Inequalities
Chapter 1, Problems 20, 21 and 22 in Spivak
20.
Prove that if
|x
-
x
0
| <
e
/2
and
|y
-
y
0
| <
e
/2,
then
|(x + y)
-
(x
0
+ y
0
)| <
e
.
Proof:
The idea is to express or estimate the expression in the
conclusion
,
|(x + y)
-
(x
0
+ y
0
)|,
in terms of expressions in the
hypothesis
,
|x
-
x
0
|
and
|y
-
y
0
| .
|(x + y)
-
(x
0
+ y
0
)|
= |(x
-
x
0
) + (y
-
y
0
)|
£
|(x
-
x
0
)| + |(y
-
y
0
)|
<
e
/2 +
e
/2 =
e
21.
Prove that if
|x
-
x
0
|
<
min
æ
è
e
2 (|y
0
| + 1)
, 1
ö
ø
,
|y
-
y
0
|
<
min
æ
è
e
2 (|x
0
| +1)
, 1
ö
ø
,
then
|x y
-
x
0
y
0
| <
e
.
Proof:
The idea again is to express or estimate the expression in the
conclusion
,
|x y
-
x
0
y
0
|,
in terms of expressions in the
hypothesis
,
|x
-
x
0
|
and
|y
-
y
0
| .
|x y
-
x
0
y
0
|
= |x (y
-
y
0
) + (x
-
x
0
) y
0
|
£
|x||y
-
y
0
| + |y
0
||x
-
x
0
|
= I + II.
I
£
|( x
-
x
0
) + x
0
| |y
-
y
0
|
£
(1 + |x
0
|)
e
2 (|x
0
| + 1)
<
e
/2.
II
£
|y
0
|
e
2 (|y
0
| +1)
<
e
/2.
22.
Show that if x
0
¹
0, and
|x
-
x
0
| <
min
æ
è
|x
0
|
2
e
2
,
|x
0
|
2
ö
ø
,
then
ê
ê
1
x
-
1
x
0
ê
ê
<
e
.
Proof:
Note that
|x|
= |x
0
+ (x
-
x
0
)|
³
|x
0
|
-
|x
0
|
2
=
|x
0
|
2
.
ê
ê
1
x
-
1
x
0
ê
ê
=
ê
ê
x
-
x
0
x x
0
ê
ê
=
1
|x|
1
|x
0
|
|x
-
x
0
|
£
2
|x
0
|
1
|x
0
|
|x
-
x
0
|
<
2
|x
0
|
1
|x
0
|
|x
0
|
2
e
2
=
e
.
File translated from T
E
X by
T
T
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On 07 Sep 2006, 16:13.