BSc Computer Science and Mathematics

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.

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.

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

• state the definition of convergent sequences and series and to verify or disprove these directly in easy examples,
• state the completeness axiom of the reals and do simple calculations with suprema and infima of bounded sets,
• calculate limits of sequences using the algebra of limits for sequences and the standard list of null sequences,
• state the monotone convergence theorem and the sandwich rule for sequences and apply them to calculate limits of sequences,
• state various convergence tests for series (e.g. comparison test or the ratio test) and use them to detect convergence or divergence of series,
• state the definition of continuous functions and verify or disprove this in easy examples,
• formulate characterizations  of continuity in terms of convergent sequences and in terms of limits of functions,
• state the intermediate value theorem and the boundedness theorem and apply them to solve equations,
• calculate limits of functions, using the algebra of limits of functions and the sandwich rule for limits of functions,
• state the definition of differentiable functions and to verify or disprove this in easy examples,
• calculate derivatives using the chain rule, the algebra of differentiable functions and the rule on derivatives of compositional inverses,
• state Rolle’s theorem, the Mean Value Theorem and L’Hpital’s Rule and to apply them to recognise the shape of functions (e.g. existence of local extrema, surjectivity of the derivative) and to calculate limits,
• prove major theorems (named above and in the online notes).

• 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.

• Coursework; Weighting within unit 20%
• End of semester examination; Weighting within unit 80%

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.

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

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.