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Introductory graduate-level analysis courses usually require an undergraduate course in real analysis or advanced calculus as a prerequisite. However, even students who have taken the appropriate prerequisites often feel inadequately prepared. The contents of the prior courses may not line up precisely with the expectations for analysis at a graduate level. Moreover, the material is challenging and lack of retention is a common issue. Finally, even students with a firm grasp on the concepts may not yet be adept at writing analysis proofs. The goal of this book is to help students bridge these gaps. In my experience of teaching first-year graduate courses on real or complex analysis, the most common issue in student preparation has to do with metric space theory. Most undergraduate analysis courses include an introduction to this topic, and so the key concepts are usually familiar. But that level of exposure often turns out to be insufficient. Metric space topology is the underlying language of mathematical analysis, and to succeed at the graduate level students need to become proficient in this language. This book originated as a set of introductory notes that I wrote for those graduate courses, to help review the core concepts and provide additional practice in analysis proof-writing. The text covers most of the standard introductory analysis material from the beginning, starting with the completeness of the real numbers. However, the focus is placed on tools and concepts that I expect to need the most review, and the treatment of certain topics is rather streamlined. For example, single-variable calculus is covered only briefly in the final chapter, as a means to show off how certain metric space concepts and tools apply to this context. The book is intended for self-study, with exercises fully incorporated into the text. These exercises are often used to develop important results which serve as the basis for subsequent proofs and exercises. A complete set of solutions is provided at the end of the book, but I strongly recommended that readers take the time to think through these problems carefully and write out their own proofs before making use of the solutions. Simply reading proofs or looking up solutions contributes little to long-term understanding and retention. Proficiency in the language of analysis is built through active engagement, by puzzling over the concepts and working through strategies for proofs.