MAT3213

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	Sets:Definitions Sets:Operations Functions:Definition Functions:Domain Functions:Range Functions:Injective Functions:Bijective Functions:Composition Functions:Inverses Functions:Forward Image Functions:Inverse Image Functions:Restriction
Week 2	1.2 & 1.3	Natural Numbers:Well-Ordering Proofs:Induction Induction:Variants Sets:Cardinality Sets:Finite Sets:Countable Sets:Uncountable	Mathematical Proofs Sets:Operations Functions:Injective Functions:Bijective
Week 3	2.1-2.2	Real Numbers Real Numbers:Algebraic Properties Real Numbers:Order Inequalities Absolute Value	Functions:Operations
Week 4	2.3-2.4	Real Numbers:Completeness Archimedean Property Suprema of Subsets Infima of Subsets Real Numbers:Rational Real Numbers:Irrational	Real Numbers:Algebraic Properties Inequalities Real Numbers:Order Absolute Value
Week 5	2.5 & 3.1	Intervals Real Numbers:Cardinality Sequences Sequences:Convergence Sequences:Limits	Sets:Cardinality Inequalities Bounded Sets
Week 6	3.2	Sequences:Tails Sequences:Limit Laws Sequences:Bounded when Convergent Sequences:Squeeze Theorem	Sequences:Convergence Sequences:Limits
Week 7	Review Midterm exam
Week 8	3.3	Sequences:Monotone Euler's Number	Proofs:Induction Bounded Sets Sequences:Limit Laws
Week 9	3.4	Subsequences Theorem:Monotone Sequence Theorem:Bolzano-Weierstrass Sequences:Limit Superior Sequences:Limit Inferior	Sequences:Monotone Sequences:Limit Laws
Week 10	3.5-3.6	The Cauchy Criterion Sequences:Contractive Properly Divergent Sequences	Sequences:Convergence Sequences:Subsequences
Week 11	3.7	Series Series:Nth-Term Test Series:Cauchy Criterion Series:Nonnegative Terms Series:Comparison Tests	Sequences:Convergence Sequences:Limits The Cauchy Criterion
Week 12	4.1	Neighborhoods in R Cluster Points Limits of Real Functions	Sequences:Convergence Sequences:Limits The Cauchy Criterion
Week 13	4.2	The Sequential Criterion for Limits Divergence Criteria	Sequences:Convergence Sequences:Limits
Week 14	4.3	One-Sided Limits of Functions Infinite Limits of Functions Limits of Functions at Infinity	Limits of Real Functions Divergence Criteria