Rigorous Introduction to Real Analysis

Real Analysis is the rigorous foundation underlying calculus, requiring precision and proof. If you are ready to transition from computational mathematics to formal mathematical reasoning, this course provides the essential depth. Upon completing this course, you will possess a robust understanding of the logical structure of analysis, enabling you to read, construct, and verify mathematical proofs concerning functions and limits on the real line. What you'll learn: * Understand the axiomatic definition of the real numbers and the topology of the real line. * Master the formal definitions of limits, sequences, and series using rigorous epsilon-delta proof techniques. * Apply the fundamental theorems of continuous functions, including the Intermediate Value and Extreme Value Theorems. * Learn the rigorous theory behind differentiation and the Mean Value Theorem for single-variable functions. * Practice the construction of Riemann integrals and analyze key properties like the Fundamental Theorem of Calculus. * Analyze the concepts of uniform convergence and the properties of function sequences and series. The course begins with the foundational structure of the real numbers, progresses through limits and continuity, and culminates in a rigorous treatment of differentiation, integration, and convergence. This program is designed for beginners who have familiarity with basic calculus but require a deep, proof-based understanding of the underlying principles. Start building your mathematical rigor today.