Students moving on from calculus or precalculus to higher-level mathematics will find that this book, which is based on the problem-solving approach developed by Pólya, is an invaluable resource. The book begins with extensive directions on approaching mathematical definitions, examples, and theorems, and it concludes with suggested projects for independent study. In the beginning, the book provides a great deal of direction on approaching mathematical definitions, examples, and theorems.
Students will apply the four-step methodology proposed by Pólya, beginning with an analysis of the issue at hand, followed by the formulation of a solution strategy, the execution of that strategy, and finally, the interpretation of the findings. In addition to the Pólya method of proofreading, this book emphasizes reading proofs with attention and producing them effectively. The writers have provided many problems, examples, illustrations, and exercises, some of which include hints and solutions, intending to enhance the student’s capacity to understand and write proofs.
Historical connections are made throughout the work, and students are urged to use the relatively comprehensive bibliography to begin making their historical connections in addition to the connections provided throughout the narrative. This book includes chapters on sequences, convergence, and metric spaces for those who want to bridge the gap between the standard course in calculus and one in analysis. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on those topics for those who want to bridge the gap.