This is a pre-publication version of An Infinite Descent into Pure Mathematics: it is a print copy of Version 0.7, which is freely available in PDF format via the book's website: https://infinitedescent.org Although it is a pre-publication version, it is sufficient for use by students and course instructors for most undergraduate-level introductory pure mathematics courses, but some of the later sections lack exercises and the hints and solutions in the appendix are incomplete. Book description: An Infinite Descent into Pure Mathematics is an introductory pure mathematics textbook with a focus on logical reasoning, proof-writing and problem-solving. Part I covers the core concepts of pure mathematics—logic, sets, functions, induction, relations and infinity—and builds a system of strategies for creating and writing mathematical proofs. Part II introduces several specific topics within pure mathematics, including number theory, combinatorics, real analysis, infinite cardinality, and probability theory. The appendices include guidance for proof-writing, hints and solutions to selected exercises, technical details that would distract from the big picture if covered in the main text, and typesetting mathematical documents using LaTeX. The reader is assumed to have a basic background in mathematics at the level of elementary algebra, arithmetic and geometry: in order to get the most out of the textbook, you should feel comfortable manipulating equations and inequalities involving variables, addition, subtraction, multiplication, division and exponentiation.