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WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

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## 1.3 The Completeness Axiom

Definition

Let ER be nonempty

(i) The set E is said to be bounded above if and only if there is an MR such that aM for all aE

(ii) A number M is called an upper bound of the set E if and only if aM for all aE

(iii) A number s is called supremum of the set E if and only if s is an upper bound of E and sM 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.)

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## 1.3 The Completeness Axiom

Definition

Let ER be nonempty

(i) The set E is said to be bounded above if and only if there is an MR such that aM for all aE

(ii) A number M is called an upper bound of the set E if and only if aM for all aE

(iii) A number s is called supremum of the set E if and only if s is an upper bound of E and sM 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.)

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## 1.3 The Completeness Axiom

Definition

Let ER be nonempty

(i) The set E is said to be bounded above if and only if there is an MR such that aM for all aE

(ii) A number M is called an upper bound of the set E if and only if aM for all aE

(iii) A number s is called supremum of the set E if and only if s is an upper bound of E and sM 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.)

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## 1.3 The Completeness Axiom

Definition

Let ER be nonempty

(i) The set E is said to be bounded above if and only if there is an MR such that aM for all aE

(ii) A number M is called an upper bound of the set E if and only if aM for all aE

(iii) A number s is called supremum of the set E if and only if s is an upper bound of E and sM 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.)

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## 1.3 The Completeness Axiom

Definition

Let ER be nonempty

(i) The set E is said to be bounded above if and only if there is an MR such that aM for all aE

(ii) A number M is called an upper bound of the set E if and only if aM for all aE

(iii) A number s is called supremum of the set E if and only if s is an upper bound of E and sM 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.)

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### Remark:

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

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### Remark:

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

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### Proof:

Let s1 and s2be suprema of the same set E. Then both s1

and s2 are upper bounds of E,whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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### Proof:

Let s1 and s2be suprema of the same set E.Then both s1

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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### Proof:

Let s1 and s2be suprema of the same set E. Then both s1

and s2 are upper bounds of E,whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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### Proof:

Let s1 and s2be suprema of the same set E. Then both s1

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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### Proof:

Let s1 and s2be suprema of the same set E. Then both s1

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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### Proof:

Let s1 and s2be suprema of the same set E. Then both s1

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1s2and s2s1. We conclude by the TrichotomyProperty that s1 =s2 2

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

sup E − ǫ <asup E

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

sup E − ǫ <asup E

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### Postulate 4:

[Completeness Axiom]

If E is a nonempty subset of R that is bounded above, then E has a (finite) supermum.

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### Postulate 4:

[Completeness Axiom]

If E is a nonempty subset of R that is bounded above, then E has a (finite) supermum.

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Theorem (Archimedean Principle)

Given positive real numbers a and b, there is an integer nN such that b<na.

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Theorem (Archimedean Principle)

Given positive real numbers a and b, there is an integer nN such that b<na.

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Theorem (Density of Rationals)

If a,bR satisfy a<b, then there is a qQ such that a <q <b.

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Theorem (Density of Rationals)

If a,bR satisfy a<b, then there is a qQ such that a <q <b.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Definition

Let ER be nonempty

(i) The set E is said to be bounded below if and only if there is an mR such that am for all aE

(ii) A number m is called an lower bound of the set E if and only if am for all aE

(iii) A number t is called infimum of the set E if and only if t is an lower bound of E and tm 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.

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Theorem

Let ER 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

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Theorem

Let ER 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

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Theorem

Let ER 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

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Theorem

Let ER 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

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Theorem (Monotone Property)

Suppose that AB are nonempty subsets of R (i) If B has a supremum, then sup Asup B (ii) If B has an infimum, then inf Ainf B

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Theorem (Monotone Property)

Suppose that AB are nonempty subsets of R (i) If B has a supremum, then sup Asup B (ii) If B has an infimum, then inf Ainf B

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Theorem (Monotone Property)

Suppose that AB are nonempty subsets of R (i) If B has a supremum, then sup Asup B (ii) If B has an infimum, then inf Ainf B

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Theorem (Monotone Property)

Suppose that AB are nonempty subsets of R (i) If B has a supremum, then sup Asup B (ii) If B has an infimum, then inf Ainf B

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## Thank you.

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