McCluskey A., McMaster B. Undergraduate Analysis: A Working Textbook

Oxford: Oxford University Press, 2018. — 397 p. — ISBN: 0198817576.Analysis underpins calculus, much as calculus underpins virtually all mathematical sciences. A sound understanding of analysis results and techniques is therefore valuable for a wide range of disciplines both within mathematics itself and beyond its traditional boundaries. This text seeks to develop such an understanding for undergraduate students on mathematics and mathematically related programmes. Keenly aware of contemporary students' diversity of motivation, background knowledge and time pressures, it consistently strives to blend beneficial aspects of the workbook, the formal teaching text, and the informal and intuitive tutorial discussion. The authors devote ample space and time for development of confidence in handling the fundamental ideas of the topic. They also focus on learning through doing, presenting a comprehensive range of examples and exercises, some worked through in full detail, some supported by sketch solutions and hints, some left open to the reader's initiative. Without undervaluing the absolute necessity of secure logical argument, they legitimise the use of informal, heuristic, even imprecise initial explorations of problems aimed at deciding how to tackle them. In this respect they authors create an atmosphere like that of an apprenticeship, in which the trainee analyst can look over the shoulder of the experienced practitioner.A Note to the Instructor.
A Note to the Student Reader.
Preliminaries.
Limit of a sequence — an idea, a definition, a tool.
Interlude: different kinds of numbers.
Up and down — increasing and decreasing sequences.
Sampling a sequence — subsequences.
Special (or specially awkward) examples.
Endless sums — a first look at series.
Continuous functions — the domain thinks that the graph is unbroken.
Limit of a function.
Epsilontics and functions.
Infinity and function limits.
Differentiation — the slope of the graph.
The Cauchy condition — sequences whose terms pack tightly together.
More about series.
Uniform continuity — continuity’s global cousin.
Differentiation — mean value theorems, power series.
Riemann integration — area under a graph.
The elementary functions revisited.
Exercises: for additional practice.
Suggestions for further reading.True PDF