Handbook

The core ideas from calculus are typically first presented informally, focussing on applying results rather than understanding why they work. Mirroring the way mathematics developed historically, this topic revisits the central ideas seen in calculus and introduces methods to formalise these. This begins with the study of mathematical proof, including techniques for constructing proofs, which will be applied throughout the topic. These methods are first used to examine numbers and sets, including key properties of the natural, rational and real numbers, the idea of a limit, and the convergence of sequences and series. The concepts developed are then used to explore properties of functions, such as continuity, and provide a framework to formalise the foundations of differential and integral calculus. Sequences and series of functions are also considered, with a focus on Taylor series. The topic also considers how the ideas developed allow for further generalisation, such as to metric spaces.