MATA172 Introduction to mathematical analysis 2 (5 cr)

The limit of a sequence and its algebraic and order properties. The connection between continuity and limits with sequences. Supremum and infimum. The monotone sequence theorem and the Bolzano Weierstrass theorem. Bolzano's theorem and other basic results concerning continuous functions. Properties of elementary functions and their inverses. Uniform continuity.

In this study-module, students practice reading and writing mathematical texts, as well as using mathematics in speech. They learn the basic skills in logical reasoning and exercise their application. The course is a continuation of the MATA171 course.

After the course, the student:

• knows the epsilon-definition of the limit of a sequence and can apply it to justify concrete limits.
• is able to utilize the basic results of sequential limits to justify the convergence of concrete sequences.
• knows the characterizations for continuity and limits of a real function using sequences.
• knows the definitions of supremum and infimum, and can determine the supremum and infimum of a given set.
• is able to apply the monotone sequence theorem and the Bolzano Weierstrass theorem.
• knows Bolzano's theorem, the boundedness of continuous functions on a closed interval, and other basic results concerning continuous functions.
• knows the basic properties of elementary functions and their inverses.
• has acquainted themselves with uniform continuity of a function.
• is able to write precise mathematical statements and proofs.
• is capable of reading mathematical texts dealing with real functions.

Introduction to mathematical analysis 1

Lecture notes (in Finnish)

The contents of the study-module correspond to

P. Fitzpatrick: Advanced Calculus, chap. 2-3.

or

Protter & Morrey: A first course in real analysis, chap. 2.5 and 3.1-3.5.