Sergey Zelik
Email: S.zelik@surrey.ac.uk
Day 
Time 
Room 
weeks 












MAT1032: There will be one class test and an examination.
Examination counts for 75% of semester marks
There will be three unassessed CourseWorks for MAT1032
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 BolzanoWeierstrass Theorem. pdf Content: Limsup, Liminf, limit points, BolzanoWeierstrass
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, onesided 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
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