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
<!-Date-> Week�1	<!-Sections-> 1.1	<!-Topics-> Basic Terminology	<!-Prerequisite Skills-> Sets of Objects	<!-Student Learning Outcomes-> Subsets The definition of equality between two sets Commonly used sets
<!-Date-> Week�1	<!-Sections-> 1.1	<!-Topics-> Set Operations	<!-Prerequisite Skills-> Basic Terminology De Morgans Laws in Logic	<!-Student Learning Outcomes-> Union, intersection and complements of sets De Morgans Laws for sets Infinite Unions and intersections of sets
<!-Date-> Week�1	<!-Sections-> 1.1	<!-Topics-> Functions (The Cartesian product definition)	<!-Prerequisite Skills-> Domain and Range of a Function Basic Terminology Set Operations	<!-Student Learning Outcomes-> The Cartesian Product Definition of a function Domain and Range in terms of the Cartesian product Transformations and Machines
<!-Date-> Week�1/2	<!-Sections-> 1.1	<!-Topics-> Direct and Inverse Images	<!-Prerequisite Skills-> Functions(The Cartesian Product Definition) Inverse Functions Set Operations	<!-Student Learning Outcomes-> Definition of the Direct Image Definition of the Inverse Image
<!-Date-> Week�1/2	<!-Sections-> 1.1	<!-Topics-> Injective and Surjective Functions	<!-Prerequisite Skills-> Functions(The Cartesian Product Definition) Direct and Inverse Images Set Operations	<!-Student Learning Outcomes-> Injective functions Surjective functions Bijective functions
<!-Date-> Week�1/2	<!-Sections-> 1.1	<!-Topics-> Inverse Functions	<!-Prerequisite Skills-> Injective and Surjective Functions Direct and Inverse Images	<!-Student Learning Outcomes-> Definition of Inverse functions Criteria for an Inverse of a function to exist
<!-Date-> Week�1/2	<!-Sections-> 1.1	<!-Topics-> Composition of Functions	<!-Prerequisite Skills-> Functions(The Cartesian Product Definition) Direct and Inverse Images	<!-Student Learning Outcomes-> Definition of a composition function When function composition is defined
<!-Date-> Week�1/2	<!-Sections-> 1.1	<!-Topics-> Restrictions on Functions	<!-Prerequisite Skills-> Domain and Range	<!-Student Learning Outcomes-> Define the restriction of a function Positive Square Root function
<!-Date-> Week�2	<!-Sections-> 1.2	<!-Topics-> Mathematical Induction	<!-Prerequisite Skills-> Basic Terminology Set Operations	<!-Student Learning Outcomes-> Well-ordering principal Principal of Mathematical induction The principal of Strong Induction
<!-Date-> Week�2	<!-Sections-> 1.3	<!-Topics-> Finite and Infinite Sets	<!-Prerequisite Skills-> Set Operations Injective and Surjective Functions	<!-Student Learning Outcomes-> Definition of finite and infinite sets Uniqueness Theorem If T is a subset of S and T is infinite, then S is also infinite.
<!-Date-> Week�2	<!-Sections-> 1.3	<!-Topics-> Countable Sets	<!-Prerequisite Skills-> Injective and Surjective Functions Finite and Infinite Sets	<!-Student Learning Outcomes-> Countable and Uncountable sets The set of rational numbers is countable Cantor's Theorem
<!-Date-> Week�3	<!-Sections-> 2.1	<!-Topics-> Algebraic Properties of the Real Numbers	<!-Prerequisite Skills-> Field Properties	<!-Student Learning Outcomes-> Algebraic properties of the Real Numbers
<!-Date-> Week�3	<!-Sections-> 2.1	<!-Topics-> Rational and Irrational Numbers	<!-Prerequisite Skills-> The square root function Functions(The Cartesian Product Definition) Definition of Even and Odd Numbers	<!-Student Learning Outcomes-> The Rational Numbers Proof that the Square Root of 2 does not exist in the rational numbers The Irrational Numbers
<!-Date-> Week�2	<!-Sections-> 2.1	<!-Topics-> The Ordering Properties of the Real Numbers	<!-Prerequisite Skills-> Inequalities Algebraic properties of the Real Numbers	<!-Student Learning Outcomes-> The ordering properties of the real numbers Tricotomy property If 0 <= a < x for each x in the positive real numbers, then a = 0.
<!-Date-> Week�2	<!-Sections-> 2.1	<!-Topics-> Inequalities	<!-Prerequisite Skills-> The Ordering Properties of the Real Numbers The Algebraic Properties of the Real Numbers	<!-Student Learning Outcomes-> Using the order properties to solve equations Arithmetic-geometric mean Bernoulli's Inequality
<!-Date-> Week�2/3	<!-Sections-> 2.2	<!-Topics-> Absolute Value and the Real Line	<!-Prerequisite Skills-> The Algebraic Properties of the Real Numbers Inequalities	<!-Student Learning Outcomes-> The absolute value function The Triangle Inequality Distance between elements of the real numbers Definition of an epsilon neighborhood
<!-Date-> Week�3	<!-Sections-> 2.3	<!-Topics-> Suprema, Infima, and the Completeness Property	<!-Prerequisite Skills-> Inequalities Absolute Value and the Real Line	<!-Student Learning Outcomes-> Upper and lower bounds of sets Definition of the suprema and infima of a set Thed completeness property of the real numbers
<!-Date-> Week�3	<!-Sections-> 2.4	<!-Topics-> Applications of the Supremum Property	<!-Prerequisite Skills-> Inequalities Absolute Value and the Real Line Suprema, Infima, and the Completeness Property	<!-Student Learning Outcomes-> Bounded Functions The Archimedean Property The existence of the square root of 2 Density of the rational numbers in the real numbers
<!-Date-> Week�3/4	<!-Sections-> 2.5	<!-Topics-> Intervals	<!-Prerequisite Skills-> Inequalities Suprema, Infima, and the Completeness Property	<!-Student Learning Outcomes-> Types of Intervals Characterization of Intervals Nested intervals The Nested Intervals Property Demonstrate that the real numbers are not countable
<!-Date-> Week�4	<!-Sections-> 3.1	<!-Topics-> Sequences and Their Limits	<!-Prerequisite Skills-> Absolute Value and the Real Line Countable Sets Applications of the Supremum Property	<!-Student Learning Outcomes-> Definition of the limit of a sequence The uniqueness of limits in the real numbers Tails of sequences Examples of common sequences
<!-Date-> Week�4	<!-Sections-> 3.2	<!-Topics-> The Limit Laws for Sequences	<!-Prerequisite Skills-> Bounded sets Sequences and Their Limits	<!-Student Learning Outcomes-> Bounded Sequences Summation, difference, products, and quotients of sequences The squeeze theorem for sequences Divergent Sequences
<!-Date-> Week�4/5	<!-Sections-> 3.3	<!-Topics-> Monotone Sequences	<!-Prerequisite Skills-> Mathematical Induction Bounded sets Bounded Sequences	<!-Student Learning Outcomes-> Increasing and Decreasing sequences The Monotone Convergence theorem Inductively defined sequences The existence of Euler's Number
<!-Date-> Week�5	<!-Sections-> 3.4	<!-Topics-> Subsequences	<!-Prerequisite Skills-> Monotone Sequences The Limit Laws for Sequences	<!-Student Learning Outcomes-> 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
<!-Date-> Week�5	<!-Sections-> 3.4	<!-Topics-> The Bolzano Weierstrass Theorem	<!-Prerequisite Skills-> Bounded Sequences Subsequences	<!-Student Learning Outcomes-> The Bolzano Weierstrass Theorem Examples using the Bolzano Weierstrass Theorem
<!-Date-> Week�5/6	<!-Sections-> 3.4	<!-Topics-> The Limit Superior and Limit Inferior	<!-Prerequisite Skills-> Bounded sets Bounded Sequences	<!-Student Learning Outcomes-> 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
<!-Date-> Week�6	<!-Sections-> 3.5	<!-Topics-> The Cauchy Criterion for Convergence	<!-Prerequisite Skills-> The Limit of a Sequence The Limit Laws for Sequences	<!-Student Learning Outcomes-> Definition of a Cauchy sequence A sequence converges if and only if it is a Cauchy sequence Contractive sequences
<!-Date-> Week�6	<!-Sections-> 3.6	<!-Topics-> Properly Divergent Sequences	<!-Prerequisite Skills-> Monotone Sequences Divergence criteria of a sequence	<!-Student Learning Outcomes-> Limits that tend to infinity Properly divergent sequences
<!-Date-> Week�6/7	<!-Sections-> 3.7	<!-Topics-> Introduction to Infinite Series	<!-Prerequisite Skills-> The Limit of a Sequence The Cauchy Criterion for Convergence	<!-Student Learning Outcomes-> 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
<!-Date-> Week�12	<!-Sections-> 4.1	<!-Topics-> Cluster Points	<!-Prerequisite Skills-> Epsilon neighborhoods The Limit of a Sequence	<!-Student Learning Outcomes-> Definition of a cluster point The cluster point as the limit of a sequence
<!-Date-> Week�12	<!-Sections-> 4.1	<!-Topics-> The Definition of the Limit of a Function	<!-Prerequisite Skills-> Cluster Point Intervals	<!-Student Learning Outcomes-> The definition of the limit of a function at a point The uniqueness of limits at cluster points Examples of limits of functions
<!-Date-> Week�12/13	<!-Sections-> 4.1	<!-Topics-> The Sequential Criterion and Divergence Criteria	<!-Prerequisite Skills-> The Limit of a Sequence The Definition of the Limit of a Function	<!-Student Learning Outcomes-> The sequential criterion for limits of functions at a point Divergence criteria for limits The signum function
<!-Date-> Week�13	<!-Sections-> 4.2	<!-Topics-> The Limit Theorems for Functions	<!-Prerequisite Skills-> The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	<!-Student Learning Outcomes-> Functions bounded on a neighborhood of a cluster point Sums, differences, products, and quotients of limits The squeeze theorem for limits of functions Examples of Limits using the limit theorems
<!-Date-> Week�14	<!-Sections-> 4.3	<!-Topics-> One Sided Limits	<!-Prerequisite Skills-> The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	<!-Student Learning Outcomes-> The definition of the right and left hand limits of a function at a point The sequential criterion for the left and right hand limits The limit of a function at a point exists if and only if its left and right hand limits are equal
<!-Date-> Week�14/15	<!-Sections-> 4.3	<!-Topics-> Infinite Limits and Limits at Infinity	<!-Prerequisite Skills-> The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria One Sided Limits	<!-Student Learning Outcomes-> The definition of an infinite limit If the function f is less than the function g on a specified domain and f tends to infinity, then g tends to infinity on this domain as well. The definition of a limit has its independent variable approaches infinity The sequential criterion for limits at infinity