Goal of the course

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Mathematics I

November 4, 2020

Mathematics I Goal of the course

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Goal of the course

Preparation for other courses — Statistics, Microeconomics, . . .

Training of logical thinking and mathematical exactness

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Goal of the course

Preparation for other courses — Statistics, Microeconomics, . . .

Training of logical thinking and mathematical exactness

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At the end of the course students should be able to understand the analysis forsequencesand functions.

compute limits and derivatives and investigate functions

understand definitions (give positive and negative examples) and theorems (explain their meaning, neccessity of the assumptions, apply them in particular situations)

understand mathematical proofs, give mathematically exact arguments

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Mathematics I

Introduction Sequences

Functions of one real variable

Mathematics I Plan

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Mathematics I

Introduction

Sequences

Mathematics I Plan

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Mathematics I

Mathematics I Plan

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Mathematics I

Mathematics I Plan

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Textbooks

H ´ajkov ´a et al: Mathematics 1 Trench: Introduction to real analysis

Ghorpade, Limaye: A course in calculus and real analysis

Zorich: Mathematical analysis I

Rudin: Principles of mathematical analysis

Mathematics I Plan

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Textbooks

H ´ajkov ´a et al: Mathematics 1

Trench: Introduction to real analysis

Mathematics I Plan

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Textbooks

Mathematics I Plan

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Textbooks

Mathematics I Plan

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Textbooks

Mathematics I Plan

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Textbooks

Mathematics I Plan

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I.1. Sets

We take a set to be a collection of definite and distinguishable objects into a coherent whole.

x ∈A. . .x is an element (or member) of the setA x ∈/ A. . .x is not a member of the setA

A⊂B. . . the setAis a subset of the setB (inclusion) A=B . . . the setsAandBhave the same elements; the following holds:A⊂B andB ⊂A

∅. . . an empty set

A∪B. . . the union of the sets AandB A∩B. . . the intersection of the sets AandB disjoint sets . . .AandB are disjoint ifA∩B =∅ A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

A1× · · · ×Am ={[a1, . . . ,am]; a1∈A1, . . . ,am ∈Am} . . . the Cartesian product

Mathematics I I. Introduction

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I.1. Sets

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I.1. Sets

x ∈A. . .x is an element (or member) of the setA

x ∈/ A. . .x is not a member of the setA

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I.1. Sets

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I.1. Sets

A⊂B. . . the setAis a subset of the setB (inclusion)

A=B . . . the setsAandBhave the same elements; the following holds:A⊂B andB ⊂A

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I.1. Sets

A⊂B. . . the setAis a subset of the setB (inclusion) A=B . . . the setsAandB have the same elements;

the following holds:A⊂B andB ⊂A

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I.1. Sets

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I.1. Sets

A∪B. . . the union of the sets AandB

A∩B. . . the intersection of the sets AandB disjoint sets . . .AandB are disjoint ifA∩B =∅ A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

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I.1. Sets

A∪B. . . the union of the sets AandB A∩B. . . the intersection of the setsAandB

disjoint sets . . .AandB are disjoint ifA∩B =∅ A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

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I.1. Sets

A∪B. . . the union of the sets AandB A∩B. . . the intersection of the setsAandB disjoint sets . . .AandB are disjoint ifA∩B =∅

A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

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I.1. Sets

A∪B. . . the union of the sets AandB A∩B. . . the intersection of the setsAandB disjoint sets . . .AandB are disjoint ifA∩B =∅ A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

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I.1. Sets

A∪B. . . the union of the sets AandB A∩B. . . the intersection of the setsAandB disjoint sets . . .AandB are disjoint ifA∩B =∅ A\B={x ∈A; x ∈/ B}. . . a difference of the setsA andB

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I.1. Sets

LetI be a non-empty set of indices and suppose we have a system of setsAα, where the indicesα run overI.

S

α∈I

Aα . . . the set of all elements belonging to at least one of the setsAα

T

α∈I

Aα . . . the set of all elements belonging to everyAα

Example.

A1∪A2∪A3is equivalent to

3

S

i=1

Ai, and also to S

i∈{1,2,3}

Ai. Infinitely many sets:A1∪A2∪A3∪. . . is equivalent to

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

A_α . . . the set of all elements belonging to at least one of the setsAα

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

Ai.

Infinitely many sets:A1∪A2∪A3∪. . . is equivalent to

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.1. Sets

S

α∈I

T

α∈I

Example.

3

S

i=1

Ai, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

Ai.

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I.2. Logic, methods of proofs

Logic

Astatement(or proposition) is a sentence which can be declared to be either true or false.

¬, alsonon. . .negation

& (also∧) . . .conjunction, logical “and”

∨. . .disjuction(alternative), logical “or”

⇒. . .implication

⇔. . .equivalence; “if and only if”

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Logic

⇒. . .implication

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Logic

⇒. . .implication

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Logic

⇒. . .implication

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Logic

⇒. . .implication

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Logic

⇒. . .implication

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Logic

⇒. . .implication

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Theorem 1 (de Morgan rules)

Let S, Aα,α∈I, where I 6=∅, be sets. Then S\[

α∈I

Aα =\

α∈I

(S\Aα) and S\\

α∈I

Aα =[

α∈I

(S\Aα).

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Example (irrationality of √ 2)

If a real numberx solves the equationx² =2, thenx is not rational.

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I.3. Number sets

Rational numbers

The set of natural numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .} The set of rational numbers

Q= p

q; p ∈Z,q ∈N

, where ^p_q¹

1 = ^p_q²

2 if and only ifp1·q2=p2·q1.

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I.3. Number sets

Rational numbers

N={1,2,3,4, . . .} The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

The set of rational numbers

Q= p

q; p ∈Z,q ∈N

, where ^p_q¹

1 = ^p_q²

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I.3. Number sets

Rational numbers

N={1,2,3,4, . . .} The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .} The set of rational numbers

Q= p

q; p ∈Z,q ∈N

, where ^p_q¹

1 = ^p_q²

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I.3. Number sets

Real numbers R

The real numbers are sometimes also called the real line.

It is the continuum of numbers where there are no gaps.

We will explain later, what ”no gap” means.

Definition.

By the set of real numbersRwe will understand a set on which there are operations ofadditionandmultiplication (denoted by+and·), and a relation ofordering(denoted by≤), such that it has the following three groups of properties.

I. The properties of addition and multiplication and their relationships.

II. The property of being ordered.

III. The infimum axiom (completion).

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I.3. Number sets

Definition

We say that the setM ⊂Risbounded from belowif there exists a numbera∈Rsuch that for eachx ∈M we have x ≥a.

Such a number ais called alower boundof the setM. Analogously we define the notions of aset

bounded from aboveand anupper bound. We say that a setM ⊂Risboundedif it is bounded from above and below.

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I.3. Number sets

Definition

We say that the setM ⊂Risbounded from belowif there exists a numbera∈Rsuch that for eachx ∈M we have x ≥a. Such a numberais called alower boundof the setM.

Analogously we define the notions of aset

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I.3. Number sets

Definition

We say that the setM ⊂Risbounded from belowif there exists a numbera∈Rsuch that for eachx ∈M we have x ≥a. Such a numberais called alower boundof the setM. Analogously we define the notions of aset bounded from aboveand anupper bound.

We say that a setM ⊂Risboundedif it is bounded from above and below.

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I.3. Number sets

Definition

We say that the setM ⊂Risbounded from belowif there exists a numbera∈Rsuch that for eachx ∈M we have x ≥a. Such a numberais called alower boundof the setM. Analogously we define the notions of aset

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I.3. Number sets

The infimum axiom:

LetM be a non-empty set bounded from below. Then there exists a unique numberg ∈Rsuch that

(i) ∀x ∈M:x ≥g,

(ii) ∀g⁰ ∈R,g⁰ >g ∃x ∈M: x <g⁰.

The numberg is denoted byinfM and is called the infimumof the setM.

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I.3. Number sets

(i) ∀x ∈M:x ≥g,

(ii) ∀g⁰ ∈R,g⁰ >g ∃x ∈M: x <g⁰.

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I.3. Number sets

(i) ∀x ∈M:x ≥g,

(ii) ∀g⁰ ∈R,g⁰ >g ∃x ∈M: x <g⁰.

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I.3. Number sets

Remark

The infimum axiom says that every non-empty set bounded from below has infimum.

The infimum of the setM is its greatest lower bound.

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I.3. Number sets

Remark

The infimum axiom says that every non-empty set bounded from below has infimum.

The infimum of the setM is its greatest lower bound.

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I.3. Number sets

Leta,b ∈R,a≤b. We denote:

Anopen interval(a,b) = {x ∈R; a<x <b}, Aclosed interval[a,b] ={x ∈R; a≤x ≤b}, Ahalf-open interval[a,b) ={x ∈R; a≤x <b}, Ahalf-open interval(a,b] ={x ∈R; a<x ≤b}.

The pointais called theleft endpoint of the interval, The pointb is called theright endpoint of the interval. A point in the interval which is not an endpoint is called aninner point of the interval.

Unbounded intervals:

(a,+∞) ={x ∈R; a<x}, (−∞,a) = {x ∈R; x <a}, analogically(−∞,a],[a,+∞)and(−∞,+∞).

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I.3. Number sets

(a,+∞) ={x ∈R; a<x}, (−∞,a) = {x ∈R; x <a}, analogically(−∞,a],[a,+∞)and(−∞,+∞).

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I.3. Number sets

(a,+∞) ={x ∈R; a<x}, (−∞,a) ={x ∈R; x <a}, analogically(−∞,a],[a,+∞)and(−∞,+∞).

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I.3. Number sets

We haveN⊂Z⊂Q⊂R. If we transfer the addition and multiplication fromRto the above sets, we obtain the usual operations on these sets.

A real number that is not rational is calledirrational. The setR\Qis called theset of irrational numbers.

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I.3. Number sets

We haveN⊂Z⊂Q⊂R. If we transfer the addition and multiplication fromRto the above sets, we obtain the usual operations on these sets.

A real number that is not rational is calledirrational. The setR\Qis called theset of irrational numbers.

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I.3. Number sets

Suprema and Maxima

Definition

LetM ⊂R. A numberG ∈Rsatisfying (i) ∀x ∈M:x ≤G,

(ii) ∀G⁰ ∈R,G⁰ <G∃x ∈M: x >G⁰, is called asupremumof the setM.

Theorem 2 (Supremum theorem)

Let M ⊂Rbe a non-empty set bounded from above. Then there exists a unique supremum of the set M.

The supremum of the setM is denoted bysupM. The following holds:supM =−inf(−M).

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I.3. Number sets

Suprema and Maxima

Definition

Theorem 2 (Supremum theorem)

The supremum of the setM is denoted bysupM. The following holds:supM =−inf(−M).

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I.3. Number sets

Suprema and Maxima

Definition

Theorem 2 (Supremum theorem)

The supremum of the setM is denoted bysupM.

The following holds:supM =−inf(−M).

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I.3. Number sets

Suprema and Maxima

Definition

Theorem 2 (Supremum theorem)

The supremum of the setM is denoted bysupM.

The following holds:supM =−inf(−M).

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I.3. Number sets

Definition

LetM ⊂R. We say thatais amaximumof the setM (denoted bymaxM) ifais an upper bound of M and a∈M. Analogously we define aminimumofM, denoted byminM.

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I.3. Number sets

Theorem 3 (Archimedean property)

For every x ∈Rthere exists n∈Nsatisfying n>x .

Theorem 4 (existence of an integer part)

For every r ∈Rthere exists aninteger part of r , i.e. a number k ∈Zsatisfying k ≤r <k +1. The integer part of r is determined uniquely and it is denoted by[r].

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I.3. Number sets

Theorem 3 (Archimedean property)

For every x ∈Rthere exists n∈Nsatisfying n>x .

Theorem 4 (existence of an integer part)

For every r ∈Rthere exists aninteger part of r , i.e. a number k ∈Zsatisfying k ≤r <k+1. The integer part of r is determined uniquely and it is denoted by[r].

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I.3. Number sets

Theorem 5 (nth root)

For every x ∈[0,+∞)and every n ∈Nthere exists a unique y ∈[0,+∞)satisfying yⁿ=x .

Theorem 6 (density of Q and R \ Q )

Let a,b ∈R, a<b. Then there exist r ∈Qsatisfying a<r <b and s ∈R\Qsatisfying a<s<b.

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I.3. Number sets

Theorem 5 (nth root)

For every x ∈[0,+∞)and every n ∈Nthere exists a unique y ∈[0,+∞)satisfying yⁿ=x .

Theorem 6 (density of Q and R \ Q )

Let a,b ∈R, a<b. Then there exist r ∈Qsatisfying a<r <b and s ∈R\Qsatisfying a<s<b.

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II.1. Introduction

II. Limit of a sequence

Definition

Suppose that to each natural numbern∈Nwe assign a real numberan. Then we say that{an}^∞_n=1is asequence of real numbers. The numberan is called thenth member of this sequence.

A sequence{an}^∞_n=1 is equal to a sequence{bn}^∞_n=1 if an=bnholds for everyn∈N.

By theset of all members of the sequence{a_n}^∞_n=1we understand the set

{x ∈R; ∃n∈N: an =x}.

Mathematics I II. Limit of a sequence

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II.1. Introduction

II. Limit of a sequence

Definition

Suppose that to each natural numbern∈Nwe assign a real numberan. Then we say that{an}^∞_n=1is asequence of real numbers.

The numberan is called thenth member of this sequence.

{x ∈R; ∃n∈N: an =x}.

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II.1. Introduction

II. Limit of a sequence

Definition

{x ∈R; ∃n∈N: an =x}.

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II.1. Introduction

II. Limit of a sequence

Definition

A sequence{an}^∞_n=1 is equal to a sequence{bn}^∞_n=1 if an =bnholds for everyn∈N.

{x ∈R; ∃n∈N: an =x}.

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II.1. Introduction

II. Limit of a sequence

Definition

A sequence{an}^∞_n=1 is equal to a sequence{bn}^∞_n=1 if an =bnholds for everyn∈N.

{x ∈R; ∃n∈N: an=x}.

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II.1. Introduction

Posloupnost {1/n}

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II.1. Introduction

Posloupnost {(–1)^n}

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II.1. Introduction

Posloupnost {n}

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II.1. Introduction

Posloupnost {P_n}

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II.1. Introduction

Definition

We say that a sequence{an}is

bounded from aboveif the set of all members of this sequence is bounded from above,

bounded from belowif the set of all members of this sequence is bounded from below,

boundedif the set of all members of this sequence is bounded.

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II.1. Introduction

Definition

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II.1. Introduction

Definition

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II.1. Introduction

Definition

We say that a sequence{a_n}is

increasingifan <an+1 for everyn ∈N,

decreasingifan >an+1 for everyn ∈N, non-decreasingifan ≤an+1 for everyn∈N, non-increasingifan ≥an+1 for everyn∈N. A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

increasingifan <an+1 for everyn ∈N, decreasingifan >an+1 for everyn ∈N,

non-decreasingifan ≤an+1 for everyn∈N, non-increasingifan ≥an+1 for everyn∈N. A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

increasingifan <an+1 for everyn ∈N, decreasingifan >an+1 for everyn ∈N, non-decreasingifan ≤an+1 for everyn ∈N,

non-increasingifan ≥an+1 for everyn∈N. A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

increasingifan <an+1 for everyn ∈N, decreasingifan >an+1 for everyn ∈N, non-decreasingifan ≤an+1 for everyn ∈N, non-increasingifan ≥an+1 for everyn ∈N.

A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

increasingifan <an+1 for everyn ∈N, decreasingifan >an+1 for everyn ∈N, non-decreasingifan ≤an+1 for everyn ∈N, non-increasingifan ≥an+1 for everyn ∈N. A sequence{an}ismonotoneif it satisfies one of the conditions above.

A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

increasingifan <an+1 for everyn ∈N, decreasingifan >an+1 for everyn ∈N, non-decreasingifan ≤an+1 for everyn ∈N, non-increasingifan ≥an+1 for everyn ∈N. A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{a_n}isstrictly monotoneif it is increasing or decreasing.

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II.1. Introduction

Definition

Let{a_n}and{b_n}be sequences of real numbers.

By thesum of sequences{an}and{bn}we understand a sequence{a_n+bn}.

Analogously we define adifferenceand aproduct of sequences.

Suppose all the members of the sequence{b_n}are non-zero. Then by thequotient of sequences{an} and{b_n}we understand a sequence{^a_bⁿ

n}.

Ifλ∈R, then by theλ-multiple of the sequence{an} we understand a sequence{λa_n}.

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II.1. Introduction

Definition

Suppose all the members of the sequence{b_n}are non-zero. Then by thequotient of sequences{an} and{b_n}we understand a sequence{^a_bⁿ

n}.

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II.1. Introduction

Definition

Suppose all the members of the sequence{b_n}are non-zero. Then by thequotient of sequences{an} and{bn}we understand a sequence{^a_bⁿ

n}.

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II.1. Introduction

Definition

Suppose all the members of the sequence{b_n}are non-zero. Then by thequotient of sequences{an} and{bn}we understand a sequence{^a_bⁿ

n}.

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II.2. Convergence of sequences

Definition

We say that a sequence{a_n}has alimitwhich equals to a numberA∈Rif to every positive real numberεthere exists a natural numbern0such that for every index n≥n0we have|a_n−A|< ε, i.e.

∀ε ∈R, ε >0∃n0∈N∀n ∈N,n ≥n0: |an−A|< ε.

We say that a sequence{a_n}isconvergentif there exists A∈Rwhich is a limit of{an}.

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Definition

We say that a sequence{a_n}has alimitwhich equals to a numberA∈Rif to every positive real numberεthere exists a natural numbern0such that for every index n≥n0we have|a_n−A|< ε, i.e.

∀ε ∈R, ε >0∃n0∈N∀n ∈N,n ≥n0: |an−A|< ε.

We say that a sequence{a_n}isconvergentif there exists A∈Rwhich is a limit of{a_n}.

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ε a-ε a+a

n0

(98)

ε a-ε a+a

n0

ε a-

ε a+

(99)

n0

ε a-ε a+a

n0

ε a-

ε a+

(100)

n0

ε a-ε a+a

n0

ε a-

ε a+

(101)

ε a-ε a+a

ε a-

ε a+

n0

(102)

n0

ε a-ε a+a

ε a-

ε a+

n0

(103)

n0

ε a-ε a+a

ε a-

ε a+

n0

(104)

ε a-ε a+a

n0

ε a-

ε a+

(105)

n0

ε a-ε a+a

n0

ε a-

ε a+

(106)

n0

ε a-ε a+a

n0

ε a-

ε a+

(107)

Theorem 7 (uniqueness of a limit)

Every sequence has at most one limit.

We use the notation lim

n→∞an=Aor simplyliman=A.

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Theorem 7 (uniqueness of a limit)

Every sequence has at most one limit.

We use the notation lim

n→∞an=Aor simplyliman=A.

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ε A-ε A+

n0

B

A

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ε A-ε A+

n0

B

A B-ε B+ε

A-ε A+ε

(111)

ε A-ε A+

n0

B

A B-ε B+ε

A-ε A+ε

n1

(112)

ε A-ε A+

n0 n₂

B

A B-ε B+ε

A-ε A+ε

n1

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Remark

Let{a_n}be a sequence of real numbers andA∈R. Then liman =A⇔lim(an−A) =0⇔lim|an−A|=0.

Theorem 8

Every convergent sequence is bounded.

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Remark

Let{a_n}be a sequence of real numbers andA∈R. Then liman =A⇔lim(an−A) =0⇔lim|an−A|=0.

Theorem 8

Every convergent sequence is bounded.

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n0

a–1 a+1a

(116)

n0

a–1 a+1

a–1 a+1a

(117)

n0

a–1 a+1

a–1 a+1a

(118)

n0

a–1 a+1

a–1 a+1a

(119)

a–1 a+1

n0

a–1 a+1a

(120)

a–1 a+1

n0

m M

n0

a–1 a+1a