This text is designed for students preparing to engage in their first struggles to understand and write proofs and to read mathematics independently. Intended for use as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology, the book teaches in detail how to construct examples and non-examples to help understand a new theorem or definition; it shows how to discover the outline of a proof in the form of the theorem and how logical structures determine the forms that proofs may take. The text is meant to be used interactively, frequently asking the reader to pause and work on an example or a problem before continuing, and encouraging the student to engage the topic at hand and to learn from failed attempts at solving problems. The book may also be used as the main text for a "transitions" course bridging the gap between calculus and higher mathematics. The material that one can assume from any introductory course in mathematics is used for the (numerous) exercises. The book begins by showing how to use examples, counterexamples, and extremes to test definitions and theorems. It then turns to proofs, first using an informal langage to elucidate the formal structure, and then discussing the formal language of mathematical logic. The book concludes with a set of "Laboratories" in which the student can practice the skills learned in the earlier chapters on topics in set theory and function theory.