Kaisa_2012_3_photo by Veikko Somerpuro

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11.2.2020 kl. 08:00 - 2.3.2020 kl. 23:59



Programme: Bachelor's programme in Science.

Module: Basics studies in Mathematics.

The course is optional within the programme but compulsory for the mathematics study track.

The course is available to the students of other programmes upon agreement with the teacher.

Calculus 1A: Limits and Differentiation and Calculus 1B: Integration are prerequisite for taking this course. The course can be taken simultaneously with Calculus IB.


This is proof based calculus or real analysis course. Concepts familiar from earlier Calculus courses, such as continuity are defined precisely. Axioms of the real number system, especially the completeness of the ordering of the real numbers are studied.

Students will learn to prove statements about sets of real numbers or real valued functions based on exact definitions, with various proof techniques, such as estimating quantities with triangle inequality, so called epsilon-delta method, using induction, and proof by contradiction.

Within mathematics study track, first year of studies.

In other study tracks, according to schedule of track/programme.

Course covers the following main topics

  1. Axioms of real numbers
  2. Completeness, supremum and infimum
  3. Convergence of sequences
  4. Proofs of Bolzano-Weirstrass theorem, extreme value theorem, intermediate value theorem among others
  5. Continuity rigorously
  6. Uniform continuity
  7. Pointwise and uniform convergence of series of functions

Either by final exam or combination of class starters, exercises and final exam.

All necessary study material can be found in the course area. An online free book, Trench: Elementary Real Analysis will be also used.


Scale 1-5 of grades will be used. Points consist of either final exam 100% or 30% exercises, 20% class starters, 50% final exam. Passing course with grade 1 will require approximately half of the points of the exam and for the best grade approximately 5/6 of the exam points are required.

The course is offered every year in the spring (period III).

Prof. Tuomo Kuusi