MAT1032: Real Analysis I


Sergey Zelik


Class Details






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

Additional reading

External links

A brilliant engine for the online computing of partial derivatives, integrals, solving min-max problems and for doing many other things!