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Undergraduate Analysis


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Table of Contents

1: Preliminaries 2: Limit of a sequence, an idea, a definition, a tool 3: Interlude: different kinds of numbers 4: Up and down - increasing and decreasing sequences 5: Sampling a sequence - subsequences 6: Special (or specially awkward) examples 7: Endless sums - a first look at series 8: Continuous functions - the domain thinks that the graph is unbroken 9: Limit of a function 10: Epsilontics and functions 11: Infinity and function limits 12: Differentiation - the slope of the graph 13: The Cauchy condition - sequences whose terms pack tightly together 14: More about series 15: Uniform continuity - continuity's global cousin 16: Differentiation - mean value theorems, power series 17: Riemann integration - area under a graph 18: The elementary functions revisited

About the Author

Brian McMaster studied at Queen's University Belfast, graduating with a PhD in 1972, and has served his alma mater department in various capacities including those of Adviser of Studies, Head of Research and Associate Director of Education. His publication profile covers over sixty refereed journal articles, mostly in the area of analytic topology but incorporating a smattering of applications in disciplines as diverse as probabilistic metric spaces and decision support theory. He has successfully supervised twelve individual postgraduate programmes including eight PhDs. He is presently formally retired but continues to deliver a full undergraduate teaching load on a voluntary basis, thus witnessing his lifelong commitment to and passion for communicating mathematics to students. His teaching interests focus around analysis (real and complex) and set theory and their development into various fields especially that of analytic topology. Aisling McCluskey graduated from Queens University Belfast with a PhD in 1990 and subsequently was awarded a postdoctoral fellowship in Toronto. She was appointed to a permanent lectureship in Mathematics in NUI, Galway in January 1992. Since then, she has established a meaningful and rewarding academic career there, maintaining an active research profile whilst holding the teaching and learning of mathematics central to her academic endeavour. She has enjoyed numerous international visiting researcher positions, and in like kind has hosted many eminent researchers at NUI, Galway. She has received institutional and national awards for excellence in teaching. In scholarly pursuit of her passion for and commitment to mathematics education, she completed a postgraduate certificate in teaching and learning in higher education (2009), a postgraduate diploma (2010) and a Masters degree (2011) at NUI Galway.


The clear, concise writing makes this book ideal for equipping undergraduates with a solid conceptual framework for approaching analysis rigorously and confidently. * V.K. Chellamuthu, CHOICE *

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