Mathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education the traditional development of analysis, often rather divorced from the calculus which they learnt at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus at school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate the new ideas are related to school topics and are used to extend the reader's understanding of those topics. A first course in analysis at college is always regarded as one of the hardest in the curriculum. However, in this book the reader is led carefully through every step in such a way that he/she will soon be predicting the next step for him/herself. In this way the subject is developed naturally: students will end up not only understanding analysis, but also enjoying it.
Table of Contents
Firm foundations 1 (23)
Why start with numbers?
How far can we get with primary school
What extra assumption is needed?
Gradually getting there 24 (51)
Why do we need sequences?
How do they behave?
How do they help us work out infinite
A functional approach 75 (68)
What do all the familiar functions mean?
How do they behave?
How are they related?
Calculus at last 143(46)
How do we work out gradients?
How does that lead to differentiation?
How does that help us to find averages and
An integrated conclusion 189(32)
How do we define the area under a graph?
Why should that have anything to do with
Solutions to exercises 221(68)