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Online course materials for MATH20111

Real Analysis

Unit code: MATH20111
Credit Rating: 10
Unit level: Level 2
Teaching period(s): Semester 1
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

Level 2 students only


The course unit unit aims to introduce the basic concepts of limit and convergence (of real sequences, series and functions) and to indicate how these are treated rigorously, and then show how these ideas are used in the development of real analysis.


The first part of the course discusses the convergence of real sequences and series.

The second part of the course discusses the concept of limit for real-valued functions of a real variable. This concept is then used to define and investigate the concepts of continuity and differentiability for such functions.

Learning outcomes

On completion of this unit successful students will be able to:

  • evaluate the limits of a wide class of real sequences;
  • determine whether or not real series are convergent by comparison with standard series or using the Ratio Test;
  • understand the concept of continuity and be familiar with the statements and some proofs of the standard results about continuous real functions;
  • understand the concept of the differentiability of a real valued function and be familiar with the statements of the standard results about differentiable real functions.

Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Coursework; Weighting within unit 20%
  • 2 hours end of semester examination; Weighting within unit 80%


  • Sequences. Null sequences and the standard list of null sequences. Convergent sequences, the Algebra of Limits, divergent sequences, monotone bounded sequences
  • Series. Convergent and divergent series, geometric series and the harmonic series. Series with non-negative terms, the Comparison Test, the Limit Comparison Test, the Ratio Test.
  • Continuity. Limits of real functions, sums, products and quotients of limits. Continuity of real functions, sums, products and quotients of continuous functions, the composition of continuous functions. The standard results about continuous real functions: the Intermediate Value Theorem and the Boundedness Theorem.
  • Differentiability. Differentiability of real-valued functions, sums, products and quotients of differentiable functions, Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem.

Recommended reading

Self contained course notes will be provided. A variety of textbooks on Real Analysis may be found on the unit's homepage http://personalpages.manchester.ac.uk/staff/Marcus.Tressl/teaching/RealAnalysis/index.php

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Marcus Tressl - Unit coordinator

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