This lively introductory text exposes the student to the rewards of
a rigorous study of functions of a real variable. In each chapter,
informal discussions of questions that give analysis its inherent
fascination are followed by precise, but not overly formal,
developments of the techniques needed to make sense of them. By
focusing on the unifying themes of approximation and the resolution
of paradoxes that arise in the transition from the finite to the
infinite, the text turns what could be a daunting cascade of
definitions and theorems into a coherent and engaging progression
of ideas. Acutely aware of the need for rigor, the student is much
better prepared to understand what constitutes a proper
mathematical proof and how to write one. Fifteen years of classroom
experience with the first edition of Understanding Analysis have
solidified and refined the central narrative of the second edition.
Roughly 150 new exercises join a selection of the best exercises
from the first edition, and three more project-style sections have
been added. Investigations of Euler's computation of (2), the
Weierstrass Approximation Theorem, and the gamma function are now
among the book's cohort of seminal results serving as motivation
and payoff for the beginning student to master the methods of
analysis.
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