An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis, “An Introduction to Proof through Real Analysis” is based on course material developed and refined over thirty years by Professor Daniel J. Madden. The book aims to help students learn the fundamental techniques of proof writing, starting with natural numbers and progressing to logic, set theory, topology, and continuity. It justifies the need for a rigorous definition of real numbers and culminates in the Intermediate Value Theorem, which validates the sufficiency of real numbers for solving all geometric problems. Written in an engaging narrative style, this text serves as both a complete guide for first proofs and first analysis courses.