Real Analysis module (MA32001)

Real Analysis shows you what lies beneath familiar ideas such as limits, continuity, and convergence. In this module, you will learn how these ideas are defined precisely and proved carefully.

You will develop the analytical framework that supports later study in areas such as differential equations, numerical analysis, probability, statistics, and mathematical modelling. The module begins with the real number system and the completeness property, showing how familiar results from calculus rest on precise mathematical assumptions.

You will then work with metric and normed spaces, where ideas of distance, convergence, continuity, and closed sets are developed in a more general setting. This lets you study convergence and continuity not only for numbers, but also for vectors, functions, and more abstract mathematical objects.

As the module develops, you will study compactness, connectedness, completeness, Cauchy sequences, uniform continuity, and uniform convergence. These ideas help explain when sequences converge, when functions preserve structure, and when local information leads to reliable global conclusions.

By learning to use definitions and theorems precisely, you will strengthen your ability to construct rigorous proofs, analyse abstract examples, and develop the logical thinking at the heart of pure mathematics.

What you will learn

In this module, you will:

• explore the completeness property of the real numbers
• study metric and normed spaces, which extend ideas of distance and convergence to more general mathematical spaces
• work with precise definitions of limits, continuity, closed sets, and limit points
• investigate compactness, connectedness, completeness, and Cauchy sequences
• examine uniform continuity, uniform convergence, and equicontinuity
• use major results such as Banach’s contraction mapping theorem and Arzelà-Ascoli to analyse convergence and sequences of functions
• develop the precision and confidence to construct rigorous proofs in pure mathematics.

By the end of this module, you will be able to:

• use precise definitions to analyse mathematical examples
• distinguish between different types of convergence and continuity
• apply major theorems in real analysis to solve problems
• recognise connections between compactness, completeness, continuity, and convergence
• construct rigorous proofs using definitions and logical arguments
• explain how abstract ideas in analysis support later study in pure and applied mathematics

Assignments / assessments

• Coursework (20%)
• Written exam (20%)

Teaching methods / timetable

• lectures, where the main ideas, definitions, theorems, and proof techniques in real analysis are introduced and developed
• tutorials, where you will work through examples, practise constructing proofs, and solve problems individually and in groups
• weekly problem-solving, helping you build confidence with precise definitions, abstract arguments, and rigorous mathematical reasoning