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### Course Overview

Analysis provides the rigorous, foundational underpinnings of calculus. It is centred around the notion of limits: convergence within the real numbers. Related ideas, such as infinite sums (a.k.a. series) and continuity are also visited in this course.

Care is needed to properly use the delicate formal concept of limits. At the same time, limits are often intuitive, and we aim to reconcile this intuition with correct mathematical reasoning. The emphasis throughout this course is on rigorous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.

### Course Details

Study Type Level Undergraduate 2 First Sub Session 15 credits (7.5 ECTS credits) Aberdeen No Dr Alexey SevastyanovProfessor Benjamin Martin

### Qualification Prerequisites

• Programme Level 2

None.

None.

No

### Course Description

- Fundamental properties of real numbers: field operations, order, completeness.

- Sequences and limits: convergence, basic examples, methods of deducing convergence, properties of convergent sequences, the Bolzano-Weierstrass Theorem.

- Infinite sums (series): convergence, convergence tests.

- Functions of one real variable: limits and continuity, methods of deducing limits, Extreme Value Theorem, Intermediate Value Theorem, uniform continuity.

Syllabus

• Properties of the real numbers: Field operations, Order, Completeness, Density of the real numbers.
• Sequences: Convergence (epsilon-delta), Properties of limits, Monotone Convergence Criterion, Subsequences, Bolzano-Weierstrass theorem.
• Series: Partial sums, Convergence, Properties of series, Criteria and tests for convergence, decimal representation of real numbers, Absolute convergence.
• Sets of real numbers: Closed and open sets.
• Continuous functions: Limits and continuity, Basic results on continuous functions, Uniform continuity, Extreme and intermediate value theorems, Points of discontinuity.

Course Aims

To put on a sound footing many of the results, procedures, and concepts used in Calculus. It will include a discussion of fundamental properties of real numbers, sequences and limits, series, and continuity of functions. Some applications will also be given.

Details, including assessment, may be subject to change until 31 July 2022 for 1st half-session courses and 23 December 2022 for 2nd half-session courses.

### Contact Teaching Time

Information on contact teaching time is available from the course guide.

### Teaching Breakdown

• 1 Support Tutorial during University weeks 9 - 19
• 1 Tutorial during University weeks 10 - 19

Details, including assessment, may be subject to change until 31 July 2022 for 1st half-session courses and 23 December 2022 for 2nd half-session courses.

### Summative Assessments

10x Weekly multi-choice or short answer online quizzes - 1% each

3x Standard assignments - 30% each

Alternative Resit Arrangements

Resubmission of failed elements (pass marks carried forward)

### Formative Assessment

There are no assessments for this course.

### Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
ConceptualApplyBe able to use the theorems of the course in unseen situations;
FactualRememberknow about basic properties of the real numbers and what distinguishes them from the rational numbers;
ConceptualApplyHave developed the ability to prove elementary results, and be able to detect fallacious arguments;
FactualApplyBe able to state the main definitions and theorems of the course;
ConceptualApplyBe able to establish the convergence of simple sequences and series
FactualRememberknow precise definitions and basic properties of elementary functions;
FactualUnderstandbe familiar with the concepts of limits and continuity.

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