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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), continuity, and differentiation, 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.

### Summative Assessments

#### Exam

Assessment Type Weighting Summative 80
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

#### Homework

Assessment Type Weighting Summative 10
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

#### Homework

Assessment Type Weighting Summative 10
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

### Formative Assessment

There are no assessments for this course.

### Resit Assessments

#### Best of written exam (100%) or written exam (80%) with carried forward in-course assessment (20%)

Assessment Type Summative
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

### Course Learning Outcomes

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

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