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### Course Overview

Analysis provides the rigourous, 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 rigourous 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 William Turner

### 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.

Learning Objectives

By the end of the course the student should:

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

### Contact Teaching Time

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

### Teaching Breakdown

• 1 during University weeks 8 - 18
• 1 Tutorial during University weeks 9 - 18

In light of Covid-19 and the move to blended learning delivery the assessment information advertised for courses may be subject to change. All updates for first-half session courses will be actioned no later than 1700 (GMT) on 18 September 2020. All updates for second half-session courses will be actioned in advance of second half-session teaching starting. Please check back regularly for updates.

### Summative Assessments

4x assessments (assignments or online tests or the mixture of the two) (25% each)

Alternative Resit Arrangements for students taking course in Academic Year 2020/21

1x Homework Exercise (100%)

### Formative Assessment

There are no assessments for this course.

### Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
ConceptualApplyHave developed the ability to prove elementary results, and be able to detect fallacious arguments;
ConceptualApplyBe able to establish the convergence of simple sequences and series
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;
FactualApplyBe able to state the main definitions and theorems of the course;
FactualUnderstandbe familiar with the concepts of limits and continuity.
FactualRememberknow precise definitions and basic properties of elementary functions;

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