Difference between revisions of "MAT3213"

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The textbook for this course is
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'''Catalog entry''':
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Introduction to Real Analysis by Bartle and Sherbert
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 +
''Prerequisites'': MAT 1224 and MAT 3013.
 +
 
 +
''Content'': Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.
 +
 
 +
 
 +
 
 +
''Sample textbook'': Introduction to Real Analysis by Bartle and Sherbert
  
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
 
A comprehensive list of all undergraduate math courses at UTSA can be found [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/  here].
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|-   
 
|-   
  
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|Week 1
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| <!--Date--> Week 1
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
       
 
−
[[Basic Terminology]]
 
−
 
 
−
||
 
−
 
 
−
* [[Set Theory| Sets of Objects]] <!-- 3013-2.1-2.3 -->
 
−
 
 
−
||
 
−
 
 
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*Subsets
 
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* The definition of equality between two sets
 
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* Commonly used sets
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;1   
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Set Operations]]
 
−
 
 
−
||
 
−
 
 
−
*[[Basic Terminology]] <!-- 3213-1.1 -->
 
−
*[[Propositional Logic|De Morgans Laws in Logic ]] <!-- 3013-1.2 and 1.3 -->
 
−
 
 
−
||
 
−
 
 
−
* Union, intersection and complements of sets
 
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* De Morgans Laws for sets
 
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* Infinite Unions and intersections of sets
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;1
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
 
 
−
[[Functions (The Cartesian product definition)]]
 
−
 
 
−
||
 
−
 
 
−
* [[Functions|Domain and Range of a Function]] <!-- 1073-1 -->
 
−
* [[Basic Terminology]] <!-- 3213-1.1 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
−
 
 
−
||
 
−
 
 
−
* The Cartesian Product
 
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* Definition of a function
 
−
* Domain and Range in terms of the Cartesian product
 
−
* Transformations and Machines
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;1/2
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
 
 
−
[[Direct and Inverse Images]]
 
−
 
 
−
||
 
−
 
 
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
 
−
* [[Inverse Functions]] <!-- 1073-7 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of the Direct Image
 
−
* Definition of the Inverse Image
 
−
 
 
−
||
 
−
 
 
−
 
 
−
|-
 
−
 
 
  
−
|Week&nbsp;1/2 
+
|| <!--Sections-->
  
−
||
+
1.1
  
−
<div style="text-align: center;">1.1</div>
+
|| <!--Topics-->  
−
 
 
−
||
 
 
    
 
    
 +
* [[Sets:Definitions]]
 +
* [[Sets:Operations]]
 +
* [[Functions:Definition]]
 +
* [[Domain of a Function|Functions:Domain]]
 +
* [[Range of a Function|Functions:Range]]
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
 +
* [[Functions:Composition]]
 +
* [[Functions:Inverses]]
 +
* [[Functions:Forward Image]]
 +
* [[Functions:Forward Image|Functions:Inverse Image]]
 +
* [[Functions:Restriction]]
  
−
[[Injective and Surjective Functions]]
+
|| <!--Prerequisite Skills-->  
−
 
 
−
||
 
−
 
 
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
 
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
 
−
* [[Set Operations]] <!-- 3213-1.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Injective functions
 
−
* Surjective functions
 
−
* Bijective functions
 
−
 
 
−
||
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;1/2
+
| <!--Date--> Week 2
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.1</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Inverse Functions]]
 
−
 
 
−
||
 
−
 
 
−
* [[Injective and Surjective Functions]]  <!-- 3213-1.1 -->
 
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of Inverse functions
 
−
* Criteria for an Inverse of a function to exist
 
−
 
 
−
||
 
−
 
 
  
−
|-
+
|| <!--Sections-->
  
 +
1.2 & 1.3
  
−
|Week&nbsp;1/2
+
|| <!--Topics--> 
  
−
||
+
* [[Natural Numbers:Well-Ordering]]
 +
* [[Proofs:Induction]]
 +
* [[Proofs:Induction|Induction:Variants]]
 +
* [[Sets:Cardinality]]
 +
* [[Sets:Finite]]
 +
* [[Sets:Countable]]
 +
* [[Sets:Uncountable]]
  
−
<div style="text-align: center;">1.1</div>
 
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
[[Composition of Functions]]
+
* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 +
* [[Sets:Operations]] <!--- 3213-1.1 --->
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
  
−
||
 
  
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
+
|| <!--Student Learning Outcomes-->  
−
* [[Direct and Inverse Images]] <!-- 3213-1.1 -->
 
  
−
||
 
  
−
* Definition of a composition function
+
|| <!--Student Learning Outcomes-->
−
* When function composition is defined
 
−
 
 
−
||  
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;1/2
+
| <!--Date--> Week 3
  
−
||
+
|| <!--Sections-->
  
−
<div style="text-align: center;">1.1</div>
+
2.1-2.2
  
−
||
+
|| <!--Topics--> 
−
 
 
  
−
[[Restrictions on Functions]]
+
* [[Real Numbers]]
 +
* [[Real Numbers:Algebraic Properties]]
 +
* [[Real Numbers|Real Numbers:Order]]
 +
* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers:Absolute Value|Absolute Value]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* [[Functions(The Cartesian Product Definition)|Domain and Range]] <!-- 3213-1.1 -->
+
* [[Functions:Operations]] <!--- DNE (recommend Modern Algebra) --->
  
−
||
+
|| <!--Student Learning Outcomes-->
−
 
 
−
* Define the restriction of a function
 
−
* Positive Square Root function
 
−
 
 
−
||
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;2
+
| <!--Date--> Week 4
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.2</div>
 
  
−
||
+
|| <!--Sections-->
  
−
[[Mathematical Induction]]
+
2.3-2.4
  
−
||
+
|| <!--Topics-->
  
−
* [[Basic Terminology]] <!-- 3213-1.1 -->
+
* [[Real Numbers|Real Numbers:Completeness]]
−
* [[Set Operations]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Archimedean Property|Archimedean Property]]
 +
* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
 +
* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
 +
* [[Real Numbers:Rational]]
 +
* [[Real Numbers:Irrational]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Well-ordering principal
+
* [[Real Numbers:Algebraic Properties]]
−
* Principal of Mathematical induction
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* The principal of Strong Induction
+
* [[Real Numbers|Real Numbers:Order]]
 +
* [[Real Numbers:Absolute Value|Absolute Value]]
  
−
||
+
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 5
  
−
|Week&nbsp;2
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">1.3</div>
 
  
−
||
+
2.5 & 3.1
−
 
 
  
−
[[Finite and Infinite Sets]]
+
|| <!--Topics--> 
  
−
||
+
* [[Real Numbers:Intervals|Intervals]]
 +
* [[Cardinality of important sets|Real Numbers:Cardinality]]
 +
* [[Real Numbers:Sequences|Sequences]]
 +
* [[Real Numbers:Sequences|Sequences:Convergence]]
 +
* [[Sequences:Limits]]
  
−
* [[Set Operations]] <!-- 3213-1.1 -->
+
|| <!--Prerequisite Skills-->  
−
* [[Injective and Surjective Functions]]  <!-- 3213-1.1 -->
 
  
−
||
+
* [[Sets:Cardinality]]
−
 
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* Definition of finite and infinite sets
+
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
−
* Uniqueness Theorem
 
−
* If T is a subset of S and T is infinite, then S is also infinite.
 
−
 
 
−
||
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 6
  
−
|Week&nbsp;2
+
|| <!--Sections-->
  
−
||
+
3.2
  
−
<div style="text-align: center;">1.3</div>
+
|| <!--Topics-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Tails]]
−
 
+
* [[Sequences:Limits|Sequences:Limit Laws]]
−
[[Countable Sets]]
+
* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
 +
* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* [[Injective and Surjective Functions]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
−
* [[Finite and Infinite Sets]] <!-- 3213-1.3 -->
+
* [[Sequences:Limits|Sequences:Limits]]
  
−
||
+
|| <!--Student Learning Outcomes-->
−
 
 
−
* Countable and Uncountable sets
 
−
* The set of rational numbers is countable
 
−
* Cantor's Theorem
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 7
  
−
|Week&nbsp;3
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">2.1</div>
 
−
 
 
−
||
 
−
 
 
−
 
 
−
[[Algebraic Properties of the Real Numbers]]
 
  
−
||
+
* Review
 +
* Midterm exam
  
−
* '''[[Field Properties]]''' <!-- DNE (recommend Modern Algebra) -->
+
|| <!--Topics-->  
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* Algebraic properties of the Real Numbers
+
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 8
  
−
|Week&nbsp;3
+
|| <!--Sections-->
  
−
||
+
3.3
  
−
<div style="text-align: center;">2.1</div>
+
|| <!--Topics-->  
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 +
* [[Euler's Number]]
  
−
[[Rational and Irrational Numbers]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[Proofs:Induction]] <!--- 3213-1.2 --->
 +
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
 +
* [[Sequences:Limits|Sequences:Limit Laws]]
  
−
* [[Restrictions on Functions| The square root function]] <!-- 3213-1.1 -->
+
|| <!--Student Learning Outcomes-->  
−
* [[Functions(The Cartesian Product Definition)]] <!-- 3213-1.1 -->
 
−
* '''[[Definition of Even and Odd Numbers]]''' <!-- DNe (recommend Modern Algebra or MAT3013 -->
 
−
 
 
−
||
 
−
 
 
−
* The Rational Numbers
 
−
* Proof that the Square Root of 2 does not exist in the rational numbers
 
−
* The Irrational Numbers
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 9
  
−
|Week&nbsp;2 
+
|| <!--Sections-->
  
−
||
+
3.4
  
−
<div style="text-align: center;">2.1</div>
+
|| <!--Topics-->  
  
−
||
+
* [[Real Numbers:Sequences|Subsequences]]
 +
* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
 +
* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Superior]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
  
−
[[The Ordering Properties of the Real Numbers]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 +
* [[Sequences:Limits|Sequences:Limit Laws]]
  
−
* [[Solving Inequalities| Inequalities]] <!-- 1073- Mod R -->
+
|| <!--Student Learning Outcomes-->  
−
* [[Algebraic properties of the Real Numbers]] <!-- 3213-2.1 -->
 
−
 
 
−
||
 
−
 
 
−
* The ordering properties of the real numbers
 
−
* Tricotomy property
 
−
* If 0 <= a < x for each x in the positive real numbers, then a = 0.
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 10
  
−
|Week&nbsp;2   
+
|| <!--Sections-->
−
 
 
−
||
 
  
−
<div style="text-align: center;">2.1</div>
+
3.5-3.6
  
−
||
+
|| <!--Topics-->
 
    
 
    
−
[[Inequalities]]
+
* [[Real Numbers:Sequences|The Cauchy Criterion]]
−
 
+
* [[Real Numbers:Sequences|Sequences:Contractive]]
−
||
+
* [[Properly Divergent Sequences]]
−
 
 
−
* [[The Ordering Properties of the Real Numbers]] <!-- 3213-2.1 -->
 
−
* [[The Algebraic Properties of the Real Numbers]] <!-- 3213-2.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Using the order properties to solve equations
 
−
* Arithmetic-geometric mean
 
−
* Bernoulli's Inequality
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;2/3 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">2.2</div>
 
−
 
 
−
||
 
−
 
 
−
[[Absolute Value and the Real Line]]
 
−
 
 
−
||
 
−
 
 
−
* [[The Algebraic Properties of the Real Numbers]] <!-- 3213-2.1 -->
 
−
* [[Inequalities]] <!-- 3213-2.1 -->
 
−
 
 
−
||
 
−
 
 
−
* The absolute value function
 
−
* The Triangle Inequality
 
−
* Distance between elements of the real numbers
 
−
* Definition of an epsilon neighborhood
 
−
 
 
−
 
 
−
|-
 
−
 
 
  
−
|Week&nbsp;3
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
−
 
+
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
−
<div style="text-align: center;">2.3</div>
 
−
 
 
−
||
 
−
 
 
−
[[Suprema, Infima, and the Completeness Property]]
 
−
 
 
−
||
 
−
 
 
−
* [[Inequalities]] <!-- 3213-2.1 -->
 
−
* [[Absolute Value and the Real Line]] <!-- 3213-2.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Upper and lower bounds of sets
 
−
* Definition of the suprema and infima of a set
 
−
* Thed completeness property of the real numbers
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 11
  
−
|Week&nbsp;3 
+
|| <!--Sections-->  
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">2.4</div>
 
  
−
|| 
+
3.7
  
−
[[Applications of the Supremum Property]]
+
|| <!--Topics--> 
  
−
||
+
* [[Series]]
 +
* [[Series|Series:Nth-Term Test]]
 +
* [[Series|Series:Cauchy Criterion]]
 +
* [[Series|Series:Nonnegative Terms]]
 +
* [[Series|Series:Comparison Tests]]
  
−
* [[Inequalities]] <!-- 3213-2.1 -->
+
|| <!--Prerequisite Skills-->  
−
* [[Absolute Value and the Real Line]] <!-- 3213-2.2 -->
 
−
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
 
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 +
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 +
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
−
* Bounded Functions
+
|| <!--Student Learning Outcomes-->
−
* The Archimedean Property
 
−
* The existence of the square root of 2
 
−
* Density of the rational numbers in the real numbers
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 12
  
−
|Week&nbsp;3/4
+
|| <!--Sections-->
  
−
||
+
4.1
  
−
<div style="text-align: center;">2.5</div>
+
|| <!--Topics-->  
  
−
||
+
* [[Neighborhoods in R]]
 +
* [[Cluster Points]]
 +
* [[The Limit of a Function|Limits of Real Functions]]
  
−
[[Intervals]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 +
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 +
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
−
* [[Inequalities]] <!-- 3213-2.1 -->
+
|| <!--Student Learning Outcomes-->  
−
* [[Suprema, Infima, and the Completeness Property]] <!-- 3213-2.3 -->
 
−
 
 
−
||
 
−
 
 
−
* Types of Intervals
 
−
* Characterization of Intervals
 
−
* Nested intervals
 
−
* The Nested Intervals Property
 
−
* Demonstrate that the real numbers are not countable
 
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
  
−
|Week&nbsp;4
+
| <!--Date--> Week 13
  
−
||
+
|| <!--Sections-->
  
−
<div style="text-align: center;">3.1</div>
+
4.2
  
−
||  
+
|| <!--Topics--> 
  
−
[[Sequences and Their Limits]]
+
* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]]
 +
* [[Divergence Criteria]]
  
−
||
+
|| <!--Prerequisite Skills-->
  
−
* [[Basis Terminology|The Natural Numbers]] <!-- 3213-1.1 -->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* [[Mathematical Induction]] <!-- 3213-1.2 -->
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* [[Applications of the Supremum Property]] <!-- 3213-2.4 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of the limit of a sequence
 
−
* The uniqueness of limits in the real numbers
 
−
* Tails of sequences
 
−
* Examples of common sequences
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
 +
| <!--Date--> Week 14
  
−
|Week&nbsp;4 
+
|| <!--Sections-->
  
−
||
+
4.3
  
−
<div style="text-align: center;">3.2</div>
+
|| <!--Topics-->  
  
−
||
+
* [[Real Function Limits:One-Sided|One-Sided Limits of Functions]]
 +
* [[Real Function Limits:Infinite|Infinite Limits of Functions]]
 +
* [[Real Function Limits:Infinite|Limits of Functions at Infinity]]
  
−
[[The Limit Theorems]]
+
|| <!--Prerequisite Skills-->
  
−
||
+
* [[The Limit of a Function|Limits of Real Functions]]
−
 
+
* [[Divergence Criteria]]
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
 
−
* [[Sequences and Their Limits]] <!-- 3213-3.1 -->
 
−
 
 
−
||
 
−
 
 
−
* Bounded Sequences
 
−
* Summation, difference, products, and quotients of sequences
 
−
* The squeeze theorem for sequences
 
−
* Divergent Sequences
 
  
 +
|| <!--Student Learning Outcomes-->
  
 +
<!-- ------------------------------------------------------------ -->
  
 
|-
 
|-
  
−
 
+
|}
−
|Week&nbsp;4/5   
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.3</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Monotone Sequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Mathematical Induction]] <!-- 3213-1.2 -->
 
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
 
−
* [[The Limit Theorems|Bounded Sequences]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Increasing and Decreasing sequences
 
−
* The Monotone Convergence theorem
 
−
* Inductively defined sequences
 
−
* The existence of Euler's Number
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
||
 
−
 
 
−
[[The Bolzano Weierstrass Theorem]]
 
−
 
 
−
||
 
−
 
 
−
* [[The Limit Laws| Bounded Sequences]] <!-- 3213-3.2 -->
 
−
* [[Subsequences]] <!-- 3213-3.4 -->
 
−
 
 
−
||
 
−
 
 
−
* The Bolzano Weierstrass Theorem
 
−
* Examples using the Bolzano Weierstrass Theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5/6 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[The Limit Superior and Limit Inferior]]
 
−
 
 
−
||
 
−
 
 
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!-- 3213-2.3 -->
 
−
* [[The Limit Laws| Bounded Sequences]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of the limit superior and limit inferior
 
−
* Equivalent statements defining the limit superior and limit inferior
 
−
* A bounded sequence converges if and only if its limit superior equals its limit inferior
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;6 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.5</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[The Cauchy Criterion for Convergence]]
 
−
 
 
−
||
 
−
 
 
−
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Cauchy sequence
 
−
* A sequence converges if and only if it is a Cauchy sequence
 
−
* Contractive sequences
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;6 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.6</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Properly Divergent Sequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[Subsequences|Divergence criteria of a sequence]] <!-- 3213-3.4 -->
 
−
 
 
−
||
 
−
 
 
−
* limits that tend to infinity
 
−
* Properly divergent sequences
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;6/7 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.7</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Introduction to Infinite Series]]
 
−
 
 
−
||
 
−
 
 
−
* [[The Limit of a Sequence]] <!-- 3213-3.1 -->
 
−
* [[The Cauchy Criterion for Convergence]] <!-- 3213-3.5 -->
 
−
 
 
−
||
 
−
 
 
−
* Sequences of partial sums
 
−
* If a series converges, then the sequence of coefficients for that series  must converge to zero.
 
−
* Examples of common series
 
−
* Comparison tests for series
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
|Week&nbsp;5 
 
−
 
 
−
||
 
−
 
 
−
<div style="text-align: center;">3.4</div>
 
−
 
 
−
|| 
 
−
 
 
−
[[Subsequences]]
 
−
 
 
−
||
 
−
 
 
−
* [[Monotone Sequences]] <!-- 3213-3.3 -->
 
−
* [[The Limit Laws]] <!-- 3213-3.2 -->
 
−
 
 
−
||
 
−
 
 
−
* Definition of a Subsequence
 
−
* If a sequence converges to limit L, then every subsequence of that sequence also converges to L.
 
−
* Definition of a divergent Sequence
 
−
* Divergence criteria of a sequence
 
−
* Monotone subsequence theorem
 
−
 
 
−
 
 
−
|-
 

Latest revision as of 20:56, 24 March 2023

Catalog entry:

Prerequisites: MAT 1224 and MAT 3013.

Content: Axiomatic definition of real numbers, including order properties and completeness; infinite sequences and their convergence; basic notions related to series and their convergence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving.


Sample textbook: Introduction to Real Analysis by Bartle and Sherbert

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Real Analysis.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

1.1


Week 2

1.2 & 1.3





Week 3

2.1-2.2


Week 4

2.3-2.4


Week 5

2.5 & 3.1


Week 6

3.2


Week 7
  • Review
  • Midterm exam


Week 8

3.3


Week 9

3.4


Week 10

3.5-3.6


Week 11

3.7


Week 12

4.1


Week 13

4.2


Week 14

4.3


Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT3213&oldid=4985"