Advanced Calculus (I)

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Advanced Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

WEN-CHINGLIEN Advanced Calculus (I)

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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.)

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1.3 The Completeness Axiom

Definition

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1.3 The Completeness Axiom

Definition

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1.3 The Completeness Axiom

Definition

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1.3 The Completeness Axiom

Definition

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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), s1 ≤s2and s2≤s1. 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), s1 ≤s2and s2≤s1. We conclude by the TrichotomyProperty that s1 =s2 2

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Proof:

and s2 are upper bounds of E,whence by Definition 1.16(iii), s1 ≤s2and s2≤s1. We conclude by the TrichotomyProperty that s1 =s2 2

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Proof:

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1 ≤s2and s2≤s1. We conclude by the TrichotomyProperty that s1 =s2 2

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Proof:

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1 ≤s2and s2≤s1. We conclude by the TrichotomyProperty that s1 =s2 2

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Proof:

and s2 are upper bounds of E, whence by Definition 1.16(iii), s1 ≤s2and s2≤s1. 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 a∈E such that

sup E − ǫ <a≤sup 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 a∈E such that

sup E − ǫ <a≤sup 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 n ∈N such that b<na.

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Theorem (Archimedean Principle)

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

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Theorem (Density of Rationals)

If a,b ∈R satisfy a<b, then there is a q ∈Q such that a <q <b.

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Theorem (Density of Rationals)

If a,b ∈R satisfy a<b, then there is a q ∈Q such that a <q <b.

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Definition

(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.

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Definition

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Definition

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Definition

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Definition

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Definition

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Definition

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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

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Theorem

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Theorem

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Theorem

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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

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Advanced Calculus (I)