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Introduction to analysis
Introduction to analysis

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Introduction to analysis


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This free OpenLearn course, Introduction to analysis, is an extract from the Open University course M208 Pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a second level course that introduces the three main branches of pure mathematics, namely group theory, analysis and linear algebra. Proofs are a vital part of pure mathematics. Students studying M208 are expected to read through a number of proofs to improve understanding of the course material and to develop mathematical skills, including producing convincing arguments and problem solving.

Introduction to analysis consists of material from M208 Unit AA1, Numbers, and has five sections in total. You should set aside between three to four hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 5%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

This extract looks at real numbers and their properties, with a particular emphasis on inequalities, which play a crucial role in analysis. It is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the course. However, a few techniques and definitions taught in earlier units in M208 are present in the extract without explanation. 

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M208 video material (although please note that the PDF may contain references to other parts of M208). In this extract, some illustrations have also been removed due to copyright restrictions.

Regrettably, mathematical and statistical content in PDF form is not accessible using a screenreader, and you may need additional help to read these documents.

Section 1 starts by revising rational numbers and their decimal representations. Then, real numbers are introduced as infinite decimals.

Section 2 looks at rules for manipulating inequalities. You will see how to find the solution set of an inequality.

Section 3 looks at various techniques for proving inequalities.

Section 4 introduces the concept of a least upper bound, which is of great importance in analysis.

Section 5 looks at how least upper bounds can be used to define arithmetic operations on the set of real numbers.

This OpenLearn course is an adapted extract from the Open University course M208 Pure mathematics.