A First Course in Analysis [Paperback]

This text on advanced calculus discusses such topics as number systems, the extreme value problem, continuous functions, differentiation, integration and infinite series. The reader will find the focus of attention shifted from the learning and applying of computational techniques to careful reasoning from hypothesis to conclusion. The book is intended both for a terminal course and as preparation for more advanced studies in mathematics, science, engineering and computation.

The first course in Analysis, which follows calculus, along with other courses, such as differential equations and elementary linear algebra, in the curricu? lum, presents special pedagogical challenges. There is a change of stress from computational manipulation to proof. Indeed, the course can become more a course in Logic than one in Analysis. Many students, caught short by a weak command of the means of mathematical discourse and unsure of what is expected of them, what the game is, suffer bouts of a kind of mental paralysis. This text attempts to address these problems in several ways: First, we have attempted to define the game as that of inquiry, by using a form of exposition that begins with a question and proceeds to analyze, ultimately to answer it, bringing in definitions, arguments, conjectures, exam? ples, etc. , as they arise naturally in the course of a narrative discussion of the question. (The true, historical narrative is too convoluted to serve for first explanations, so no attempt at historical accuracy has been made; our narra? tives are completely contrived. ) Second, we have kept the logic informal, especially in the course of preliminary speculative discussions, where common sense and plausibility? tempered by mild skepticism-serve to energize the inquiry.Background Number Systems.- ?1. Counting: The Natural Numbers.- ?2. Measurement: The Rational Numbers.- The Axioms of Ordered Fields.- ?3. Decimal Representation. Irrationals.- I ANALYSIS.- 1 lc%