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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* [[Sets:Operations]]
 
* [[Sets:Operations]]
 
* [[Functions:Definition]]  
 
* [[Functions:Definition]]  
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* [[Functions:Domain]]  
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* [[Domain of a Function|Functions:Domain]]  
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* [[Functions:Range]]  
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* [[Range of a Function|Functions:Range]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Injective]]  
 
* [[Functions:Bijective]]  
 
* [[Functions:Bijective]]  
 
* [[Functions:Composition]]  
 
* [[Functions:Composition]]  
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* [[Functions:Inverse]]  
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* [[Functions:Inverses]]  
 
* [[Functions:Forward Image]]  
 
* [[Functions:Forward Image]]  
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* [[Functions:Inverse Image]]
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* [[Functions:Forward Image|Functions:Inverse Image]]
 
* [[Functions:Restriction]]  
 
* [[Functions:Restriction]]  
  
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|| <!--Sections-->  
 
|| <!--Sections-->  
  
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1.2
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1.2 & 1.3
  
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
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* [[NaturalNumbers:Well Ordering]]
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* [[Natural Numbers:Well-Ordering]]
 
* [[Proofs:Induction]]
 
* [[Proofs:Induction]]
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* [[Induction:Variants]]
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* [[Proofs:Induction|Induction:Variants]]
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* [[Sets:Cardinality]]
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* [[Sets:Finite]]
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* [[Sets:Countable]]
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* [[Sets:Uncountable]]
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|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
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* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 
* [[Mathematical Proofs]] <!--- 3213-1.1 --->
 
* [[Sets:Operations]] <!--- 3213-1.1 --->
 
* [[Sets:Operations]] <!--- 3213-1.1 --->
 +
* [[Functions:Injective]]
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* [[Functions:Bijective]]
 +
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
  
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<!-- ------------------------------------------------------------ -->
 
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−
|-
 
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| <!--Date--> Week 3
 
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−
|| <!--Sections-->
 
−
 
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1.3
 
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|| <!--Topics-->
 
−
 
 
−
 
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* [[Finite and Infinite Sets]]
 
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* [[Countable Sets]]
 
−
 
−
|| <!--Prerequisite Skills-->
 
−
 
−
* [[Sets:Operations]]
 
−
* [[NaturalNumbers:Well Ordering]]
 
−
* [[Proofs:Induction]]
 
−
* [[Functions:Injective]]
 
−
* [[Functions:Bijective]]
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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* [[Real Numbers]]
 
* [[Real Numbers]]
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* [[RealNumbers:Algebraic Properties]]
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* [[Real Numbers:Algebraic Properties]]
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* [[RealNumbers:Order]]
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* [[Real Numbers|Real Numbers:Order]]
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* [[RealNumbers:Inequalities|Inequalities]]
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* [[Real Numbers:Inequalities|Inequalities]]
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* [[RealNumbers:Absolute Value|Absolute Value]]
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* [[Real Numbers:Absolute Value|Absolute Value]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
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| <!--Date--> Week 3
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| <!--Date--> Week 4
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
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|| <!--Topics-->  
 
|| <!--Topics-->  
  
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* [[RealNumbers:Completeness]]
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* [[Real Numbers|Real Numbers:Completeness]]
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* [[RealNumbers:Archimedean Property|Archimedean Property]]
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* [[Real Numbers:Archimedean Property|Archimedean Property]]
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* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
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* [[Real Numbers:Suprema and Infima|Suprema of Subsets]]
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* [[RealNumbers:Infima of Subsets|Infima of Subsets]]
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* [[Real Numbers:Suprema and Infima|Infima of Subsets]]
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* [[RealNumbers:Rational]]
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* [[Real Numbers:Rational]]
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* [[RealNumbers:Irrational]]
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* [[Real Numbers:Irrational]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealNumbers:Algebraic Properties]]
+
* [[Real Numbers:Algebraic Properties]]
−
* [[RealNumbers:Inequalities|Inequalities]]
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* [[Real Numbers:Inequalities|Inequalities]]
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* [[RealNumbers:Order]]
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* [[Real Numbers|Real Numbers:Order]]
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* [[RealNumbers:Absolute Value|Absolute Value]]
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* [[Real Numbers:Absolute Value|Absolute Value]]
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
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| <!--Date--> Week 4
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
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2.5
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
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* [[RealNumbers:Intervals|Intervals]]
 
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* [[RealNumbers:Cardinality]]
 
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* [[RealNumbers:Decimal Representation|Decimal Representation]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[RealNumbers:Completeness|Completeness]] <!--- 3213-2.1 --->
 
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* [[RealNumbers:Archimedean Property|Archimedean Property]]
 
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* [[RealNumbers:Bounded Subsets|Bounded Sets]]
 
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* [[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
 
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* [[RealNumbers:Infima of Subsets|Infima of Subsets]]
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Sections-->  
 
|| <!--Sections-->  
  
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3.1
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2.5 & 3.1
  
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[RealNumbers:Sequences|Sequences]]
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* [[Real Numbers:Intervals|Intervals]]
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* [[Sequences:Tails]]
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* [[Cardinality of important sets|Real Numbers:Cardinality]]
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* [[Sequences:Convergence]]
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* [[Real Numbers:Sequences|Sequences]]
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* [[Real Numbers:Sequences|Sequences:Convergence]]
 
* [[Sequences:Limits]]
 
* [[Sequences:Limits]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealNumbers:Inequalities|Inequalities]]
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* [[Sets:Cardinality]]
−
* [[RealNumbers:Absolute Value|Absolute Value]]
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* [[Real Numbers:Inequalities|Inequalities]]
 +
* [[Real Numbers:Bounded Subsets|Bounded Sets]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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3.2
 
3.2
  
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|| <!--Topics-->  
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|| <!--Topics-->
−
 
 
  
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* [[Sequences:Limit Laws]]
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* [[Real Numbers:Sequences|Sequences:Tails]]
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* [[Sequences:Bounded when Convergent]]
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* [[Sequences:Limits|Sequences:Limit Laws]]
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* [[Sequences:Squeeze Theorem]]
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* [[Real Numbers:Sequences|Sequences:Bounded when Convergent]]
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* [[Real Numbers:Sequences|Sequences:Squeeze Theorem]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Limits]]
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* [[Real Numbers:Sequences|Sequences:Convergence]]
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* [[Sequences:Limits|Sequences:Limits]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
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* [[Sequences:Monotone]]
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* [[Real Numbers:Sequences|Sequences:Monotone]]
 
* [[Euler's Number]]
 
* [[Euler's Number]]
  
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* [[Proofs:Induction]] <!--- 3213-1.2 --->
 
* [[Proofs:Induction]] <!--- 3213-1.2 --->
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* [[RealNumbers:Bounded Subsets|Bounded Sets]]
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* [[Real Numbers:Bounded Subsets|Bounded Sets]]
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* [[Sequences:Limit Laws]]
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* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
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* [[Sequences:Subsequences|Subsequences]]
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* [[Real Numbers:Sequences|Subsequences]]
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* [[Theorem:Monotone Sequence]]
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* [[Real Numbers:Sequences|Theorem:Monotone Sequence]]
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* [[Theorem:Bolzano-Weierstrass]]
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* [[Real Numbers:Sequences|Theorem:Bolzano-Weierstrass]]
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* [[Sequences:Limit Superior]]
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* [[Real Numbers:Sequences|Sequences:Limit Superior]]
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* [[Sequences:Limit Inferior]]
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* [[Real Numbers:Sequences|Sequences:Limit Inferior]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
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* [[Sequences:Monotone]]
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* [[Real Numbers:Sequences|Sequences:Monotone]]
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* [[Sequences:Limit Laws]]
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* [[Sequences:Limits|Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Topics-->  
 
|| <!--Topics-->  
 
    
 
    
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* [[Sequences:Cauchy|The Cauchy Criterion]]
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* [[Real Numbers:Sequences|The Cauchy Criterion]]
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* [[Sequences:Contractive]]
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* [[Real Numbers:Sequences|Sequences:Contractive]]
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* [[Sequences:Divergent]]
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* [[Properly Divergent Sequences]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]]
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* [[Real Numbers:Sequences|Sequences:Convergence]]
 
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
 
* [[Sequences:Subsequences]] <!--- 3213-3.4 --->
  
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* [[Series]]
 
* [[Series]]
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* [[Series:Nth-Term Test]]
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* [[Series|Series:Nth-Term Test]]
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* [[Series:Cauchy Criterion]]
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* [[Series|Series:Cauchy Criterion]]
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* [[Series:Nonnegative Terms]]
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* [[Series|Series:Nonnegative Terms]]
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* [[Series:Comparison Tests]]
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* [[Series|Series:Comparison Tests]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* [[Sequences:Cauchy|The Cauchy Criterion]] <!--- 3213-3.5 --->
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* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[Neighborhoods in 𝐑]]
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* [[Neighborhoods in R]]
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* [[Cluster Points in 𝐑]]
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* [[Cluster Points]]
−
* [[RealFunctions:Limits|Limits of Real Functions]]
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* [[The Limit of a Function|Limits of Real Functions]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
−
* [[Sequences:Cauchy|The Cauchy Criterion]] <!--- 3213-3.5 --->
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* [[Real Numbers:Sequences|The Cauchy Criterion]] <!--- 3213-3.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
* [[RealFunctionLimits:Sequential Criterion|The Sequential Criterion for Limits]]
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* [[Real Function Limits:Sequential Criterion|The Sequential Criterion for Limits]]
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* [[RealFunctionLimits:Divergence Criteria|Divergence Criteria]]
+
* [[Divergence Criteria]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Sequences:Convergence]] <!--- 3213-3.1 --->
+
* [[Real Numbers:Sequences|Sequences:Convergence]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
 
* [[Sequences:Limits]] <!--- 3213-3.1 --->
  
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|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
* [[RealFunctionLimits:One-Sided|One-Sided Limits of Functions]]
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* [[Real Function Limits:One-Sided|One-Sided Limits of Functions]]
−
* [[RealFunctionLimits:Infinite|Infinite Limits of Functions]]
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* [[Real Function Limits:Infinite|Infinite Limits of Functions]]
−
* [[RealFunctionLimits:At Infinity|Limits of Functions at Infinity]]
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* [[Real Function Limits:Infinite|Limits of Functions at Infinity]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[RealFunctions:Limits|Limits of Real Functions]]
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* [[The Limit of a Function|Limits of Real Functions]]
 +
* [[Divergence Criteria]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|-
 
|-
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|}

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"