MAT1032: Real Analysis I

## Lecturer

Sergey Zelik

E-mail: S.zelik@surrey.ac.uk

## Class Details

 Day Time Room weeks

## Assessment

MAT1032: There will be one class test and an examination.

Examination counts for 75% of semester marks

## Unassessed Assignments

There will be three unassessed CourseWorks for MAT1032

## Lectures  (Combined by topics)

Lecture 1.   Extended Introduction   pdf  Content: Examples why proofs are important, discussion on what is proof, propositional logic, axioms, theorems, etc.

Lecture 2.   Types of Numbers and Preliminaries pdf  Content: Rational, irrational, algebraic and transcendental numbers, Decimal and binary expansions of real numbers.

Lecture 3.   Sets, Functions and Related Things pdf    Content: Sets and Operations, functions, quantifiers, Cartesian products of many sets, Axiom of Choice.

Lecture 4.   Cardinality.  pdf  Content: General concepts. Countable and Continual Sets. Cantor Theorems. Power Set. Various examples.

Lecture 5.   Real Numbers and Completeness Axiom   pdf  Content: Min, Max, Sup, Inf, Axioms of Real Numbers, discussion and simple examples.

Lecture 6.   Limits of Sequences. Preliminaries   pdf    Content:  Definitions, simple examples, algebraic properties of limits.

Lecture 7.   Monotone Convergence Theorem   pdf  Content:  Monotone convergent theorem, examples, exponent and Euler limit, harmonic series and Euler constant.

Lecture 8.   Limit Points and Bolzano-Weierstrass Theorem. pdf  Content: Limsup, Liminf, limit points, Bolzano-Weierstrass theorem, Weyl theorem and examples.

Lecture 9.   Cauchy sequences and Cauchy Theorem   pdf  Content: Definitions, Cauchy theorem, nested intervals theorem, examples.

Lecture 10. Open and Closed Sets (topology).   pdf   Content: Definitions, open sets and intervals, Cantor set, examples.

Lecture 11. Continuous functions. Part I.  pdf  Content: Continuity and sequential continuity, limits, one-sided limits, removable singularities, etc.

Lecture 12. Continuous functions. Part II.  pdf  with pictures Pic1  Pic2  Pic3  Pic4  Content:  Intermediate and Extreme value theorems, inverse functions, iterations of  functions, 1D dynamics in examples

Lecture 13. Series. Part I. Preliminaries and Basic Properties.  pdf  Content: Definitions, Abel transformation, telescoping series, examples.

Lecture 14. Series. Part II. Convergence Tests   pdf   Content: Standard convergence tests and examples.

Lecture 15. Series. Part III. Rearrangements and Riemann Theorem   pdf  Content:  Riemann theorem, rearrangements of alternative harmonic series, multiple series, lattice sums and Madelung constant.

Lecture 16. Series. Part IV. Power series   pdf  Content: Definitions, radius of convergence, differentiation and integration of power series, Abel theorem, relation with analytic functions, examples