## Advanced Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

WEN-CHINGLIEN **Advanced Calculus (I)**

## 1.3 The Completeness Axiom

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded above if and only if*
*there is an M* ∈* R such that a*≤

*M for all a*∈

*E*

*(ii) A number M is called an upper bound of the set E if*
*and only if a*≤*M for all a*∈*E*

*(iii) A number s is called supremum of the set E if and*
*only if s is an upper bound of E and s* ≤*M for all upper*
*bounds M of E. (In this case we shall say that E has a*
*supermum s and shall write s=sup E.)*

WEN-CHINGLIEN **Advanced Calculus (I)**

## 1.3 The Completeness Axiom

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded above if and only if*
*there is an M* ∈* R such that a*≤

*M for all a*∈

*E*

*(ii) A number M is called an upper bound of the set E if*
*and only if a*≤*M for all a*∈*E*

*(iii) A number s is called supremum of the set E if and*
*only if s is an upper bound of E and s* ≤*M for all upper*
*bounds M of E. (In this case we shall say that E has a*
*supermum s and shall write s=sup E.)*

WEN-CHINGLIEN **Advanced Calculus (I)**

## 1.3 The Completeness Axiom

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded above if and only if*
*there is an M* ∈* R such that a*≤

*M for all a*∈

*E*

*(ii) A number M is called an upper bound of the set E if*
*and only if a*≤*M for all a*∈*E*

*(iii) A number s is called supremum of the set E if and*
*only if s is an upper bound of E and s* ≤*M for all upper*
*bounds M of E. (In this case we shall say that E has a*
*supermum s and shall write s=sup E.)*

WEN-CHINGLIEN **Advanced Calculus (I)**

## 1.3 The Completeness Axiom

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded above if and only if*
*there is an M* ∈* R such that a*≤

*M for all a*∈

*E*

*(ii) A number M is called an upper bound of the set E if*
*and only if a*≤*M for all a*∈*E*

*(iii) A number s is called supremum of the set E if and*
*only if s is an upper bound of E and s* ≤*M for all upper*
*bounds M of E. (In this case we shall say that E has a*
*supermum s and shall write s=sup E.)*

WEN-CHINGLIEN **Advanced Calculus (I)**

## 1.3 The Completeness Axiom

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded above if and only if*
*there is an M* ∈* R such that a*≤

*M for all a*∈

*E*

*(ii) A number M is called an upper bound of the set E if*
*and only if a*≤*M for all a*∈*E*

*(iii) A number s is called supremum of the set E if and*
*only if s is an upper bound of E and s* ≤*M for all upper*
*bounds M of E. (In this case we shall say that E has a*
*supermum s and shall write s=sup E.)*

WEN-CHINGLIEN **Advanced Calculus (I)**

**Remark:**

If a set has a supremum, then it has only one supremum.

WEN-CHINGLIEN **Advanced Calculus (I)**

**Remark:**

If a set has a supremum, then it has only one supremum.

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2*be suprema of the same set E. Then both s*1

*and s*2 are upper bounds of E,whence by Definition
*1.16(iii), s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2be suprema of the same set E.*Then both s*1

*and s*2 are upper bounds of E, whence by Definition
1.16(iii), *s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2*be suprema of the same set E. Then both s*1

*and s*2 are upper bounds of E,whence by Definition
*1.16(iii), s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2*be suprema of the same set E. Then both s*1

*and s*2 are upper bounds of E, whence by Definition
1.16(iii), *s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2*be suprema of the same set E. Then both s*1

*and s*2 are upper bounds of E, whence by Definition
*1.16(iii), s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*Let s*1 *and s*2*be suprema of the same set E. Then both s*1

*and s*2 are upper bounds of E, whence by Definition
*1.16(iii), s*1 ≤*s*2*and s*2≤*s*1. We conclude by the
*TrichotomyProperty that s*1 =*s*2 2

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Approximation Property For Suprema)
*If E has a supremum and* ǫ >*0 is any positive number,*
*then there is a point a*∈*E such that*

*sup E* − ǫ <*a*≤*sup E*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Approximation Property For Suprema)
*If E has a supremum and* ǫ >*0 is any positive number,*
*then there is a point a*∈*E such that*

*sup E* − ǫ <*a*≤*sup E*

WEN-CHINGLIEN **Advanced Calculus (I)**

**Postulate 4:**

[Completeness Axiom]
**If E is a nonempty subset of R that is bounded above,***then E has a (finite) supermum.*

WEN-CHINGLIEN **Advanced Calculus (I)**

**Postulate 4:**

[Completeness Axiom]
**If E is a nonempty subset of R that is bounded above,***then E has a (finite) supermum.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Archimedean Principle)

*Given positive real numbers a and b, there is an integer*
*n* ∈*N such that b*<*na.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Archimedean Principle)

*Given positive real numbers a and b, there is an integer*
*n* ∈*N such that b*<*na.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Density of Rationals)

*If a,b* ∈* R satisfy a*<

*b, then there is a q*∈

*Q such that*

*a*<

*q*<

*b.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Density of Rationals)

*If a,b* ∈* R satisfy a*<

*b, then there is a q*∈

*Q such that*

*a*<

*q*<

*b.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Definition

*Let E* ⊂**R be nonempty**

*(i) The set E is said to be bounded below if and only if*
*there is an m* ∈* R such that a*≥

*m for all a*∈

*E*

*(ii) A number m is called an lower bound of the set E if*
*and only if a*≥*m for all a* ∈*E*

*(iii) A number t is called infimum of the set E if and only if t*
*is an lower bound of E and t* ≤*m for all lower bounds m*
*of E. (In this case we shall say that E has an infimum t*
*and shall write t=inf E.)*

*(iv) E is said to be bounded if and only if it is bounded*
above and below.

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Let E*⊂**R be nonempty**

*(i) E has a supremum if and only if -E has an infimum, in*
*which case*

inf(−E) = −*sup E*

*(ii) E has an infimum if and only if -E has a supremum, in*
*which case*

sup(−E) = −*inf E*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Let E*⊂**R be nonempty**

*(i) E has a supremum if and only if -E has an infimum, in*
*which case*

inf(−E) = −*sup E*

*(ii) E has an infimum if and only if -E has a supremum, in*
*which case*

sup(−E) = −*inf E*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Let E*⊂**R be nonempty**

*(i) E has a supremum if and only if -E has an infimum, in*
*which case*

inf(−E) = −*sup E*

*(ii) E has an infimum if and only if -E has a supremum, in*
*which case*

sup(−E) = −*inf E*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Let E*⊂**R be nonempty**

*(i) E has a supremum if and only if -E has an infimum, in*
*which case*

inf(−E) = −*sup E*

*(ii) E has an infimum if and only if -E has a supremum, in*
*which case*

sup(−E) = −*inf E*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Monotone Property)

*Suppose that A*⊆**B are nonempty subsets of R***(i) If B has a supremum, then sup A*≤*sup B*
*(ii) If B has an infimum, then inf A*≥*inf B*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Monotone Property)

*Suppose that A*⊆**B are nonempty subsets of R***(i) If B has a supremum, then sup A*≤*sup B*
*(ii) If B has an infimum, then inf A*≥*inf B*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Monotone Property)

*Suppose that A*⊆**B are nonempty subsets of R***(i) If B has a supremum, then sup A*≤*sup B*
*(ii) If B has an infimum, then inf A*≥*inf B*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Monotone Property)

*Suppose that A*⊆**B are nonempty subsets of R***(i) If B has a supremum, then sup A*≤*sup B*
*(ii) If B has an infimum, then inf A*≥*inf B*

WEN-CHINGLIEN **Advanced Calculus (I)**

*Thank you.*

WEN-CHINGLIEN **Advanced Calculus (I)**