Elementary Real Analysis is a core course in nearly all college and university mathematics departments. It enables students to develop a deep understanding of the key concepts of calculus from a mature perspective. Elements of Real Analysis is a student-friendly guide to learning all the important ideas of elementary real analysis, based on the author’s many years of experience teaching the subject to typical undergraduates. It avoids the compact style of professional mathematics writing, in favor of a style that feels more comfortable to students encountering the subject for the first time. It presents topics in ways that are most easily understood, without sacrificing rigor or coverage. Important concepts are discussed in greater detail than customary in a text at this level. Using this text, students discover that real analysis is completely deducible from the axioms of the real number system. They learn the powerful techniques of limits of sequences as the primary entry to the concepts of analysis, and see the ubiquitous role sequences play in virtually all later topics. They become comfortable with topological ideas, and see how these concepts help unify the subject. They develop a unified understanding of limits, continuity, differentiability, Riemann integrability, and infinite series of numbers and functions. Students encounter many interesting examples, including “pathological” ones, that motivate the subject and help fix the concepts.