Absolute Value and the Real Line - MATH 464/506, Real...
Transcript of Absolute Value and the Real Line - MATH 464/506, Real...
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
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
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
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
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
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
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
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
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
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
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
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
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
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