MAT4213

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	5.1	Continuous Functions	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	The definition of a continuous function Sequential criterion for continuity Discontinuity criterion
Week 1/2	5.2	Combinations of Continuous Functions	Continuous Functions Composition of Functions	Sums, differences, products and quotients of continuous functions on the same domain Composition of continuous functions
Week 2	5.3	Continuous Functions on Intervals	Domain and Range of a Function Intervals Continuous Functions	Bounded Functions The boundedness theorem on closed and bounded intervals Definitions of absolute maximum and absolute minimum of a function The maximum-minimum theorem
Week 2/3	5.3	The Intermediate Value Theorem	Continuous Functions on Intervals Continuous Functions	The Location of Roots theorem The Intermediate Value Theorem The image of a continuous functions on a closed and bounded interval is a closed and bounded interval
Week 3	5.4	Uniform Continuity	Continuous Functions Continuous Functions on Intervals	The definition of uniform continuity Nonuniform continuity criteria Uniform continuity theorem
Week 3/4	5.4	Lipschitz Functions	Uniform Continuity Functions(The Cartesian Product Definition)	Definition of a Lipschitz function If a function is Lipschtiz, then it is uniformly continuous
Week 4	5.4	The Continuous Extension Theorem	Continuous Functions on Intervals Uniform Continuity	Uniform continuity and Cauchy sequences The Continuous Extension Theorem
Week 4/5	5.4	Approximations of Continuous Functions	Domain and Range Continuous Functions on Intervals The Continuous Extension Theorem	Definition of a step function On closed and bounded intervals, continuous functions can be approximated by piecewise linear functions Weierstrass approximation theorem
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 Laws for Sequences	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 for Sequences	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 for Sequences	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 12	4.1	Cluster Points	Epsilon neighborhoods The Limit of a Sequence	Definition of a cluster point The cluster point as the limit of a sequence
Week 12	4.1	The Definition of the Limit of a Function	Cluster Point Intervals	The definition of the limit of a function at a point The uniqueness of limits at cluster points Examples of limits of functions
Week 12/13	4.1	The Sequential Criterion and Divergence Criteria	The Limit of a Sequence The Definition of the Limit of a Function	The sequential criterion for limits of functions at a point Divergence criteria for limits The signum function
Week 13	4.2	The Limit Theorems for Functions	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	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
Week 14	4.3	One Sided Limits	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	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
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	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