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	Basic Terminology	Sets of Objects
Week 1	1.1	Set Operations	Basic Terminology De Morgans Laws in Logic
Week 1	1.1	Functions (The Cartesian product definition)	Domain and Range of a Function Basic Terminology Set Operations
Week 1/2	1.1	Direct and Inverse Images	Functions(The Cartesian Product Definition) Inverse Functions Set Operations
Week 1/2	1.1	Injective and Surjective Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images Set Operations
Week 1/2	1.1	Inverse Functions	Injective and Surjective Functions Direct and Inverse Images
Week 1/2	1.1	Composition of Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images
Week 1/2	1.1	Restrictions on Functions	Domain and Range
Week 2	1.2	Mathematical Induction	Basic Terminology Set Operations
Week 2	1.3	Finite and Infinite Sets	Set Operations Injective and Surjective Functions
Week 2	1.3	Countable Sets	Injective and Surjective Functions Finite and Infinite Sets
Week 3	2.1	Algebraic Properties of the Real Numbers	Field Properties
Week 3	2.1	Rational and Irrational Numbers	The square root function Functions(The Cartesian Product Definition) Definition of Even and Odd Numbers
Week 2	2.1	The Ordering Properties of the Real Numbers	Inequalities Algebraic properties of the Real Numbers
Week 2	2.1	Inequalities	The Ordering Properties of the Real Numbers The Algebraic Properties of the Real Numbers
Week 2/3	2.2	Absolute Value and the Real Line	The Algebraic Properties of the Real Numbers Inequalities
Week 3	2.3	Suprema, Infima, and the Completeness Property	Inequalities Absolute Value and the Real Line
Week 3	2.4	Applications of the Supremum Property	Inequalities Absolute Value and the Real Line Suprema, Infima, and the Completeness Property
Week 3/4	2.5	Intervals	Inequalities Suprema, Infima, and the Completeness Property
Week 4	3.1	Sequences and Their Limits	Absolute Value and the Real Line Countable Sets Applications of the Supremum Property
Week 4	3.2	The Limit Laws for Sequences	Bounded sets Sequences and Their Limits
Week 4/5	3.3	Monotone Sequences	Mathematical Induction Bounded sets Bounded Sequences
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws for Sequences
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