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:ForwardImage Functions:InverseImage Functions:Restriction
Week 2	1.2	NaturalNumbers:Well Ordering Proofs:Induction Induction:Variants	Mathematical Proofs Sets:Operations
Week 3	1.3	Finite and Infinite Sets Countable Sets	Sets:Operations NaturalNumbers:Well Ordering Proofs:Induction Functions:Injective Functions:Bijective
Week 3	2.1-2.2	Real Numbers RealNumbers:Algebraic Properties RealNumbers:Order Inequalities Absolute Value	Functions:Operations
Week 3	2.3-2.4	RealNumbers:Completeness Archimedean Property Suprema of Subsets Infima of Subsets Suprema of Subsets RealNumbers:Rational RealNumbers:Irrational	RealNumbers:Algebraic Properties Inequalities RealNumbers:Order Absolute Value
Week 4	2.5	Intervals RealNumbers:Cardinality Decimal Representation	Completeness Archimedean Property Bounded Sets Suprema of Subsets Infima of Subsets
Week 5	3.1	Sequences Sequences:Tails Sequences:Limits	Inequalities Absolute Value
Week 6	3.2	Sequences:Limit Laws	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 Subsequence Theorem:Bolzano-Weierstrass	Sequences:Monotone Sequences:Limit Laws
Week 5	3.4	The Bolzano Weierstrass Theorem	Bounded Sequences Subsequences
Week 5/6	3.4	The Limit Superior and Limit Inferior	Bounded sets Bounded Sequences
Week 6	3.5	The Cauchy Criterion for Convergence	The Limit of a Sequence The Limit Laws for Sequences
Week 6	3.6	Properly Divergent Sequences	Monotone Sequences Divergence criteria of a sequence
Week 6/7	3.7	Introduction to Infinite Series	The Limit of a Sequence The Cauchy Criterion for Convergence
Week 12	4.1	Cluster Points	Epsilon neighborhoods The Limit of a Sequence
Week 12	4.1	The Definition of the Limit of a Function	Cluster Point Intervals
Week 12/13	4.1	The Sequential Criterion and Divergence Criteria	The Limit of a Sequence The Definition of the Limit of a Function
Week 13	4.2	The Limit Theorems for Functions	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria
Week 14	4.3	One Sided Limits	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria
Week 14/15	4.3	Infinite Limits and Limits at Infinity	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria One Sided Limits