Aguilar C. An Introduction to Real Analysis 2022

Textbook in PDF format

Preliminaries
Sets, Numbers, and Proofs
Mathematical Induction
Functions
Countability
The Real Numbers
Introduction
Algebraic and Order Properties
The Absolute Value
The Completeness Axiom
Applications of the Supremum
Nested Interval Theorem
Sequences
Limits of Sequences
Limit Theorems
Monotone Sequences
Bolzano-Weierstrass Theorem
limsup and liminf
Cauchy Sequences
Infinite Series
Limits of Functions
Limits of Functions
Limit Theorems
Continuity
Continuous Functions
Combinations of Continuous Functions
Continuity on Closed Intervals
Uniform Continuity
Differentiation
The Derivative
The Mean Value Theorem
Taylor's Theorem
Riemann Integration
The Riemann Integral
Riemann Integrable Functions
The Fundamental Theorem of Calculus
Riemann-Lebesgue Theorem
Sequences of Functions
Pointwise Convergence
Uniform Convergence
Properties of Uniform Convergence
Infinite Series of Functions
Metric Spaces
Metric Spaces
Sequences and Limits
Continuity
Completeness
Compactness
Fourier Series
Multivariable Differential Calculus
Differentiation
Differentiation Rules and the MVT
The Space of Linear Maps
Solutions to Differential Equations
High-Order Derivatives
Taylor's Theorem
The Inverse Function Theorem