Introduction to Real Analysis: Foundations of Calculus

Are you ready to move beyond computational calculus and explore the rigorous foundations of modern mathematics? Real Analysis provides the necessary tools for understanding why calculus works, focusing on precision and formal proof techniques. By the end of this course, you will have a deep, formal understanding of the real number system and the fundamental concepts of limits, continuity, and convergence. You will gain proficiency in constructing rigorous mathematical proofs, a crucial skill for advanced study in mathematics, engineering, and theoretical computer science. What you'll learn: * Understand the axiomatic structure of the real number system, including completeness and order properties. * Master the construction and application of epsilon-delta proofs for defining limits and continuity. * Analyze the behavior of infinite sequences and series, including tests for convergence and divergence. * Learn the formal, rigorous definitions of differentiation and Riemann integration on intervals. * Practice constructing clear, logical, and rigorous mathematical arguments for foundational theorems in analysis. The course begins with foundational set theory and the structure of real numbers, progressing systematically through limits, continuity, and convergence. Subsequent sections apply these concepts to build the rigorous theory of differentiation and integration. This course is designed for absolute beginners in higher mathematics, including students and professionals who have a basic working knowledge of calculus but want to understand the underlying theory. No prior experience with formal proof writing is required. Start building your foundation in rigorous mathematical reasoning today.