Free course

Introduction to analysis

This free course is an introduction to analysis which looks at real numbers and their properties, with a particular emphasis on inequalities. 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 and finding 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. Section 5 looks at how least upper bounds can be used to define arithmetic operations on the set of real numbers.

Course learning outcomes

After studying this course, you should be able to:

  • find a rational and an irrational number between any two distinct real numbers
  • solve inequalities by rearranging into simpler equivalent forms and solve inequalities involving modulus signs
  • state and use the Triangle Inequality and use the Binomial Theorem and mathematical induction to prove inequalities which involve an integer n
  • use the strategies introduced for determining least upper bounds and greatest lower bounds
  • explain how the least upper bound property is used to define arithmetical operations with real numbers and explain the meaning of rational powers.

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Course content