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	Subsets The definition of equality between two sets Commonly used sets
Week 1	1.1	Set Operations	Basic Terminology De Morgans Laws in Logic	Union, intersection and complements of sets De Morgans Laws for sets Infinite Unions and intersections of sets
Week 1	1.1	Functions (The Cartesian product definition)	Domain and Range of a Function Basic Terminology Set Operations	The Cartesian Product Definition of a function Domain and Range in terms of the Cartesian product Transformations and Machines
Week 1/2	1.1	Direct and Inverse Images	Functions(The Cartesian Product Definition) Inverse Functions Set Operations	Definition of the Direct Image Definition of the Inverse Image
Week 1/2	1.1	Injective and Surjective Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images Set Operations	Injective functions Surjective functions Bijective functions
Week 1/2	1.1	Inverse Functions	Injective and Surjective Functions Direct and Inverse Images	Definition of Inverse functions Criteria for an Inverse of a function to exist
Week 1/2	1.1	Composition of Functions	Functions(The Cartesian Product Definition) Direct and Inverse Images	Definition of a composition function When function composition is defined
Week 1/2	1.1	Restrictions on Functions	Domain and Range	Define the restriction of a function Positive Square Root function
Week 2	1.2	Mathematical Induction	Basic Terminology Set Operations	Well-ordering principal Principal of Mathematical induction The principal of Strong Induction
Week 2	1.3	Finite and Infinite Sets	Set Operations Injective and Surjective Functions	Definition of finite and infinite sets Uniqueness Theorem If T is a subset of S and T is infinite, then S is also infinite.
Week 2	1.3	Countable Sets	Injective and Surjective Functions Finite and Infinite Sets	Countable and Uncountable sets The set of rational numbers is countable Cantor's Theorem
Week 3	2.1	Algebraic Properties of the Real Numbers	Field Properties	Algebraic properties of the Real Numbers
Week 3	2.1	Rational and Irrational Numbers	The square root function Functions(The Cartesian Product Definition) Definition of Even and Odd Numbers	The Rational Numbers Proof that the Square Root of 2 does not exist in the rational numbers The Irrational Numbers
Week 2	2.1	The Ordering Properties of the Real Numbers	Inequalities Algebraic properties of the Real Numbers	The ordering properties of the real numbers Tricotomy property If 0 <= a < x for each x in the positive real numbers, then a = 0.
Week 2	2.1	Inequalities	The Ordering Properties of the Real Numbers The Algebraic Properties of the Real Numbers	Using the order properties to solve equations Arithmetic-geometric mean Bernoulli's Inequality
Week 2/3	2.2	Absolute Value and the Real Line	The Algebraic Properties of the Real Numbers Inequalities	The absolute value function The Triangle Inequality Distance between elements of the real numbers Definition of an epsilon neighborhood
Week 3	2.3	Suprema, Infima, and the Completeness Property	Inequalities Absolute Value and the Real Line	Upper and lower bounds of sets Definition of the suprema and infima of a set Thed completeness property of the real numbers
Week 3	2.4	Applications of the Supremum Property	Inequalities Absolute Value and the Real Line Suprema, Infima, and the Completeness Property	Bounded Functions The Archimedean Property The existence of the square root of 2 Density of the rational numbers in the real numbers
Week 3/4	2.5	Intervals	Inequalities Suprema, Infima, and the Completeness Property	Types of Intervals Characterization of Intervals Nested intervals The Nested Intervals Property Demonstrate that the real numbers are not countable
Week 4	3.1	Sequences and Their Limits	The Natural Numbers Mathematical Induction Applications of the Supremum Property	Definition of the limit of a sequence The uniqueness of limits in the real numbers Tails of sequences Examples of common sequences
Week 4	3.2	The Limit Theorems	Bounded sets Sequences and Their Limits	Bounded Sequences Summation, difference, products, and quotients of sequences The squeeze theorem for sequences Divergent Sequences
Week 4/5	3.3	Monotone Sequences	Mathematical Induction Bounded sets Bounded Sequences	Increasing and Decreasing sequences The Monotone Convergence theorem Inductively defined sequences The existence of Euler's Number
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem
Week 5	3.4	The Bolzano Weierstrass Theorem	Bounded Sequences Subsequences	The Bolzano Weierstrass Theorem Examples using the Bolzano Weierstrass Theorem
Week 5/6	3.4	The Limit Superior and Limit Inferior	Bounded sets Bounded Sequences	Definition of the limit superior and limit inferior Equivalent statements defining the limit superior and limit inferior A bounded sequence converges if and only if its limit superior equals its limit inferior
Week 6	3.5	The Cauchy Criterion for Convergence	The Limit of a Sequence The Limit Laws	Definition of a Cauchy sequence A sequence converges if and only if it is a Cauchy sequence Contractive sequences
Week 6	3.6	Properly Divergent Sequences	Monotone Sequences Divergence criteria of a sequence	limits that tend to infinity Properly divergent sequences
Week 6/7	3.7	Introduction to Infinite Series	The Limit of a Sequence The Cauchy Criterion for Convergence	Sequences of partial sums If a series converges, then the sequence of coefficients for that series must converge to zero. Examples of common series Comparison tests for series
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem
Week 5	3.4	Subsequences	Monotone Sequences The Limit Laws	Definition of a Subsequence If a sequence converges to limit L, then every subsequence of that sequence also converges to L. Definition of a divergent Sequence Divergence criteria of a sequence Monotone subsequence theorem