MAT3213

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