absolute value and the real line - math 464/506, real...
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Absolute Value and the Real LineMATH 464/506, Real Analysis
J. Robert Buchanan
Department of Mathematics
Summer 2007
J. Robert Buchanan Absolute Value and the Real Line
![Page 2: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/2.jpg)
Absolute Value
Definition
The absolute value of a real number a, denoted by |a|, isdefined by
|a| =
a if a > 0,0 if a = 0,−a if a < 0.
Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.
Proof.
J. Robert Buchanan Absolute Value and the Real Line
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Absolute Value
Definition
The absolute value of a real number a, denoted by |a|, isdefined by
|a| =
a if a > 0,0 if a = 0,−a if a < 0.
Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.
Proof.
J. Robert Buchanan Absolute Value and the Real Line
![Page 4: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/4.jpg)
Absolute Value
Definition
The absolute value of a real number a, denoted by |a|, isdefined by
|a| =
a if a > 0,0 if a = 0,−a if a < 0.
Theorem1 |ab| = |a||b| for all a, b ∈ R.2 |a|2 = a2 for all a ∈ R.3 If c ≥ 0, then |a| ≤ c if and only if −c ≤ a ≤ c.4 −|a| ≤ a ≤ |a| for all a ∈ R.
Proof.
J. Robert Buchanan Absolute Value and the Real Line
![Page 5: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/5.jpg)
Triangle Inequality
Theorem
If a, b ∈ R, then |a + b| ≤ |a| + |b|.
Proof.
Corollary
If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.
Corollary
If a1, a2, . . . , an are any real numbers, then
|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.
J. Robert Buchanan Absolute Value and the Real Line
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Triangle Inequality
Theorem
If a, b ∈ R, then |a + b| ≤ |a| + |b|.
Proof.
Corollary
If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.
Corollary
If a1, a2, . . . , an are any real numbers, then
|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.
J. Robert Buchanan Absolute Value and the Real Line
![Page 7: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/7.jpg)
Triangle Inequality
Theorem
If a, b ∈ R, then |a + b| ≤ |a| + |b|.
Proof.
Corollary
If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.
Corollary
If a1, a2, . . . , an are any real numbers, then
|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.
J. Robert Buchanan Absolute Value and the Real Line
![Page 8: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/8.jpg)
Triangle Inequality
Theorem
If a, b ∈ R, then |a + b| ≤ |a| + |b|.
Proof.
Corollary
If a, b ∈ R, then1 ||a| − |b|| ≤ |a − b|,2 |a − b| ≤ |a| + |b|.
Corollary
If a1, a2, . . . , an are any real numbers, then
|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|.
J. Robert Buchanan Absolute Value and the Real Line
![Page 9: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/9.jpg)
Real Line
Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.
Definition
Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set
Vǫ(a) = {x ∈ R : |x − a| < ǫ}.
Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:
−ǫ < x − a < ǫ
a − ǫ < x < a + ǫ
J. Robert Buchanan Absolute Value and the Real Line
![Page 10: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/10.jpg)
Real Line
Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.
Definition
Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set
Vǫ(a) = {x ∈ R : |x − a| < ǫ}.
Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:
−ǫ < x − a < ǫ
a − ǫ < x < a + ǫ
J. Robert Buchanan Absolute Value and the Real Line
![Page 11: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/11.jpg)
Real Line
Remarks:Geometrically we may regard |a| as the distance along thenumber line from 0 to a.The distance between a and b in R is |a − b|.
Definition
Let a ∈ R and ǫ > 0. The ǫ-neighborhood of a is the set
Vǫ(a) = {x ∈ R : |x − a| < ǫ}.
Remark: x ∈ Vǫ(a) means x satisfies the following equivalentinequalities:
−ǫ < x − a < ǫ
a − ǫ < x < a + ǫ
J. Robert Buchanan Absolute Value and the Real Line
![Page 12: Absolute Value and the Real Line - MATH 464/506, Real …banach.millersville.edu/~bob/math464/AbsoluteValue/main.pdfAbsolute Value and the Real Line MATH 464/506, Real Analysis J](https://reader030.vdocument.in/reader030/viewer/2022040515/5e6f2fb8bdf4280d70462b31/html5/thumbnails/12.jpg)
Result
Theorem
Let a ∈ R. If x belongs to the neighborhood Vǫ(a) for everyǫ > 0, then x = a.
Proof.
J. Robert Buchanan Absolute Value and the Real Line
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Result
Theorem
Let a ∈ R. If x belongs to the neighborhood Vǫ(a) for everyǫ > 0, then x = a.
Proof.
J. Robert Buchanan Absolute Value and the Real Line
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Homework
Read Section 2.2.
Page 34: 1, 2, 14 , 15
Boxed problems should be written up separately and submittedfor grading at class time on Friday.
J. Robert Buchanan Absolute Value and the Real Line