Difference between revisions of "MAT3213"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Rewrite up to Week 8.)
 
(41 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
The textbook for this course is
+
'''Catalog entry''':
−
Introduction to Real Analysis by Bartle and Sherbert
+
 
 +
''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].
Line 23: Line 30:
 
* [[Sets:Operations]]
 
* [[Sets:Operations]]
 
* [[Functions:Definition]]  
 
* [[Functions:Definition]]  
−
* [[Functions:Domain]]  
+
* [[Domain of a Function|Functions:Domain]]  
−
* [[Functions:Range]]  
+
* [[Range of a Function|Functions:Range]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Bijective]]  
 
* [[Functions:Bijective]]  
 
* [[Functions:Composition]]  
 
* [[Functions:Composition]]  
−
* [[Functions:Inverse]]  
+
* [[Functions:Inverses]]  
−
* [[Functions:ForwardImage]]  
+
* [[Functions:Forward Image]]  
−
* [[Functions:InverseImage]]
+
* [[Functions:Forward Image|Functions:Inverse Image]]
 
* [[Functions:Restriction]]  
 
* [[Functions:Restriction]]  
  
Line 46: Line 53:
 
|| <!--Sections-->  
 
|| <!--Sections-->  
  
−
1.2
+
1.2 & 1.3
  
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[NaturalNumbers:Well Ordering]]
+
* [[Natural Numbers:Well-Ordering]]
 
* [[Proofs:Induction]]
 
* [[Proofs:Induction]]
−
* [[Induction:Variants]]
+
* [[Proofs:Induction|Induction:Variants]]
 +
* [[Sets:Cardinality]]
 +
* [[Sets:Finite]]
 +
* [[Sets:Countable]]
 +
* [[Sets:Uncountable]]
 +
 
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
Line 58: Line 70:
 
* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 
* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 
* [[Sets:Operations]] <!--- 3213-1.1 --->
 
* [[Sets:Operations]] <!--- 3213-1.1 --->
 +
* [[Functions:Injective]]
 +
* [[Functions:Bijective]]
 +
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
−
<!-- ------------------------------------------------------------ -->
 
−
 
−
|-
 
−
 
−
| <!--Date--> Week 3
 
−
 
−
|| <!--Sections-->
 
−
 
−
1.3
 
−
 
−
|| <!--Topics-->
 
−
 
 
−
 
−
* [[Finite and Infinite Sets]]
 
−
* [[Countable Sets]]
 
−
 
−
|| <!--Prerequisite Skills-->
 
−
 
−
* [[Sets:Operations]]
 
−
* [[NaturalNumbers:Well Ordering]]
 
−
* [[Proofs:Induction]]
 
−
* [[Functions:Injective]]
 
−
* [[Functions:Bijective]]
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 101: Line 93:
  
 
* [[Real Numbers]]
 
* [[Real Numbers]]
−
* [[RealNumbers:Algebraic Properties]]
+
* [[Real Numbers:Algebraic Properties]]
−
* [[RealNumbers:Order]]
+
* [[Real Numbers|Real Numbers:Order]]
−
* [[RealNumbers:Inequalities|Inequalities]]
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* [[RealNumbers:Absolute Value|Absolute Value]]
+
* [[Real Numbers:Absolute Value|Absolute Value]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
Line 117: Line 109:
  
  
−
| <!--Date--> Week 3
+
| <!--Date--> Week 4
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
Line 125: Line 117:
 
|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[RealNumbers:Completeness]]
+
* [[Real Numbers|Real Numbers:Completeness]]
−
* [[RealNumbers:Archimedean Property|Archimedean Property]]
+
* [[Real Numbers:Archimedean Property|Archimedean Property]]
−
* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
+
* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
−
* [[RealNumbers:Infima of Subsets|Infima of Subsets]]
+
* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
−
* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
+
* [[Real Numbers:Rational]]
−
* [[RealNumbers:Rational]]
+
* [[Real Numbers:Irrational]]
−
* [[RealNumbers:Irrational]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[RealNumbers:Algebraic Properties]]
 
−
* [[RealNumbers:Inequalities|Inequalities]]
 
−
* [[RealNumbers:Order]]
 
−
* [[RealNumbers:Absolute Value|Absolute Value]]
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
| <!--Date--> Week 4
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
2.5
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[RealNumbers:Intervals|Intervals]]
 
−
* [[RealNumbers:Cardinality]]
 
−
* [[RealNumbers:Decimal Representation|Decimal Representation]]
 
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealNumbers:Completeness|Completeness]] <!--- 3213-2.1 --->
+
* [[Real Numbers:Algebraic Properties]]
−
* [[RealNumbers:Archimedean Property|Archimedean Property]]
+
* [[Real Numbers:Inequalities|Inequalities]]
−
* [[RealNumbers:Bounded Subsets|Bounded Sets]]
+
* [[Real Numbers|Real Numbers:Order]]
−
* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
+
* [[Real Numbers:Absolute Value|Absolute Value]]
−
* [[RealNumbers:Infima of Subsets|Infima of Subsets]]
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 176: Line 141:
 
|| <!--Sections-->  
 
|| <!--Sections-->  
  
−
3.1
+
2.5 & 3.1
  
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[RealNumbers:Sequences|Sequences]]
+
* [[Real Numbers:Intervals|Intervals]]
−
* [[Sequences:Tails]]
+
* [[Cardinality of important sets|Real Numbers:Cardinality]]
 +
* [[Real Numbers:Sequences|Sequences]]
 +
* [[Real Numbers:Sequences|Sequences:Convergence]]
 
* [[Sequences:Limits]]
 
* [[Sequences:Limits]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealNumbers:Inequalities|Inequalities]]
+
* [[Sets:Cardinality]]
−
* [[RealNumbers:Absolute Value|Absolute Value]]
+
* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 201: Line 169:
 
3.2
 
3.2
  
−
|| <!--Topics-->  
+
|| <!--Topics-->
  
−
* [[Sequences:Limit Laws]]
+
* [[Real Numbers:Sequences|Sequences:Tails]]
 +
* [[Sequences:Limits|Sequences:Limit Laws]]
 +
* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
 +
* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Limits]]
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
 +
* [[Sequences:Limits|Sequences:Limits]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 240: Line 212:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Sequences:Monotone]]
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
 
* [[Euler's Number]]
 
* [[Euler's Number]]
  
Line 246: Line 218:
  
 
* [[Proofs:Induction]] <!--- 3213-1.2 --->
 
* [[Proofs:Induction]] <!--- 3213-1.2 --->
−
* [[RealNumbers:Bounded Subsets|Bounded Sets]]
+
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
−
* [[Sequences:Limit Laws]]
+
* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 263: Line 235:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Sequences:Subsequences|Subsequences]]
+
* [[Real Numbers:Sequences|Subsequences]]
−
* [[Theorem:Monotone Subsequence]]
+
* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
−
* [[Theorem:Bolzano-Weierstrass]]
+
* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Superior]]
 +
* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Monotone]]
+
* [[Real Numbers:Sequences|Sequences:Monotone]]
−
* [[Sequences:Limit Laws]]
+
* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 278: Line 252:
 
|-
 
|-
  
−
| <!--Date--> Week 5 
+
| <!--Date--> Week 10
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
  
−
3.4
+
3.5-3.6
  
 
|| <!--Topics-->  
 
|| <!--Topics-->  
 
    
 
    
−
* [[The Bolzano Weierstrass Theorem]]
+
* [[Real Numbers:Sequences|The Cauchy Criterion]]
−
 
+
* [[Real Numbers:Sequences|Sequences:Contractive]]
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Limit Laws for Sequences| Bounded Sequences]] <!--- 3213-3.2 --->
 
−
* [[Subsequences]] <!--- 3213-3.4 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 5/6 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
3.4
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[The Limit Superior and Limit Inferior]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!--- 3213-2.3 --->
 
−
* [[The Limit Laws for Sequences| Bounded Sequences]] <!--- 3213-3.2 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 6 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
3.5
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[The Cauchy Criterion for Convergence]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Limit of a Sequence]] <!--- 3213-3.1 --->
 
−
* [[The Limit Laws for Sequences]] <!--- 3213-3.2 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 6 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
3.6
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
 
* [[Properly Divergent Sequences]]
 
* [[Properly Divergent Sequences]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Monotone Sequences]] <!--- 3213-3.3 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]]
−
* [[Subsequences|Divergence criteria of a sequence]] <!--- 3213-3.4 --->
+
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 365: Line 275:
 
|-
 
|-
  
−
 
+
| <!--Date--> Week 11
−
| <!--Date--> Week 6/7 
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
Line 374: Line 283:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[Introduction to Infinite Series]]
+
* [[Series]]
 +
* [[Series|Series:Nth-Term Test]]
 +
* [[Series|Series:Cauchy Criterion]]
 +
* [[Series|Series:Nonnegative Terms]]
 +
* [[Series|Series:Comparison Tests]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[The Limit of a Sequence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* [[The Cauchy Criterion for Convergence]] <!--- 3213-3.5 --->
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 +
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 387: Line 301:
 
|-
 
|-
  
−
 
+
| <!--Date--> Week 12
−
| <!--Date--> Week 12  
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
Line 394: Line 307:
 
4.1
 
4.1
  
−
|| <!--Topics-->  
+
|| <!--Topics-->  
  
 +
* [[Neighborhoods in R]]
 
* [[Cluster Points]]
 
* [[Cluster Points]]
 +
* [[The Limit of a Function|Limits of Real Functions]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Absolute Value and the Real Line|Epsilon neighborhoods]] <!--- 3213-2.2 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* [[The Limit of a Sequence]] <!--- 3213-3.1 --->
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
 
+
* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 12 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
4.1
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[The Definition of the Limit of a Function]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[Cluster Point]] <!--- 3213-4.1 --->
 
−
* [[RealNumbers:Intervals|Intervals]] <!--- 3213-2.5 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 12/13 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
4.1
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[The Sequential Criterion and Divergence Criteria]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Limit of a Sequence]] <!--- 3213-3.1 --->
 
−
* [[The Definition of the Limit of a Function]] <!--- 3213-4.1 --->
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 454: Line 326:
  
  
−
| <!--Date--> Week 13  
+
| <!--Date--> Week 13
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
Line 462: Line 334:
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[The Limit Theorems for Functions]]
+
* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]]
 +
* [[Divergence Criteria]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[The Definition of the Limit of a Function]] <!--- 3213-4.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
−
* [[The Sequential Criterion and Divergence Criteria]]<!--- 3213-4.1 --->
+
* [[Sequences:Limits]] <!--- 3213-3.1 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 475: Line 348:
 
|-
 
|-
  
−
 
+
| <!--Date--> Week 14
−
| <!--Date--> Week 14  
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
Line 482: Line 354:
 
4.3
 
4.3
  
−
|| <!--Topics-->  
+
|| <!--Topics-->  
  
−
* [[One Sided Limits]]
+
* [[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]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[The Definition of the Limit of a Function]] <!--- 3213-4.1 --->
+
* [[The Limit of a Function|Limits of Real Functions]]
−
* [[The Sequential Criterion and Divergence Criteria]]<!--- 3213-4.1 --->
+
* [[Divergence Criteria]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
Line 497: Line 371:
 
|-
 
|-
  
−
| <!--Date--> Week 14/15
+
|}
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
4.3
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
* [[Infinite Limits and Limits at Infinity]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Definition of the Limit of a Function]] <!--- 3213-4.1 --->
 
−
* [[The Sequential Criterion and Divergence Criteria]]<!--- 3213-4.1 --->
 
−
* [[One Sided Limits]] <!--- 3213-4.1 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 

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"