Aims: Analysis is the area of mathematics that - among other things - places calculus (differentiation and integration) on a solid foundation. While the Analysis 2 module discusses calculus, Analysis 1 introduces the key ideas and will get you used to producing rigorous arguments.

You will have seen at school how the tangent to a curve as the result of drawing two points on the curve ever closer together; but how do we talk about this in a precise way that then enables us to prove something about its properties?

Content: Analysis 1 covers three main topics, with the first two closely related.

1. Sequences and limits.

We start off by discussing how to define "limits" properly. What does it mean to say that \(1/n\to0\) as \(n\to\infty\)? With a simple example like this it is easy to have an intuitive idea, but how do we understand the statement that \((1+\frac{1}{n})^n\to{\rm e}\) as \(n\to\infty\) or prove that \(n^{1/n}\to1\) as \(n\to\infty\)? Once we want to prove more interesting things like these, we need a proper definition of what a limit really is.

2. Sums of series.

A very natural way in which such a limiting process arises is if we try to sum up infinite series. What does it mean to say that

\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =\sum_{n=1}^\infty 2^{-n}=1\)?

This you can see by thinking about cutting a cake in half repeatedly, so once more your intuition will give you an idea. But what about

\(1+\frac{1}{4}+\frac{1}{9}+\cdots=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}\)?

The first step to proving this (the proof is not easy!) is to understand what such a statement actually means.

3. Continuous functions.

What does it mean for a function to be continuous? (We will first, of course, have to think a little more carefully about what a "function" actually is - although by the time we treat functions in Analysis 1, you will have covered them in Foundations.)

After reading the above, it should be fairly clear that "something you can draw without taking your pencil off the page" is not going to get us very far - certainly not if we want to prove anything!

We will give a formal definition of what it means for a function to be continuous and then use this to prove some basic (and extremely useful) properties of continuous functions. Sometimes the results we prove might seem to be "obvious", but try proving the Intermediate Value Theorem below without having a proper definition of what it means to be continuous. ("But it's just obvious!" is not a proof.)

Let \(f\colon [a,b]\to{\mathbb R}\) be continuous, and assume that \(f(a)<f(b)\). Then for any \(c\) with \(f(a)<c<f(b)\) there exists a point \(x\in(a,b)\) such that \(f(x)=c\).

 Books:
M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises)
K.G. Binmore, Mathematical Analysis: A Straightforward Approach, CUP (1982)
L. Alcock, How To Think About Analysis, Oxford University Press (2014)