Introduction

Analysis I is an introduction to the mathematical field of Analysis. This section of the course will be held in English (a Slovenian section is available separately). Topics include: Number systems, sequences and limits, series and convergence, and limits and continuity of functions of one real variable.

Prerequisite

I'll assume that you have a reasonably strong high school mathematics background. It'd be helpful to be simultaneously enrolled in (or to have finished) Discrete Mathematics I or Theoretical Computer Science I.

Textbook and literature

There will be no set textbook adopted for the class. I will give out my handwritten notes, and carefully talk you through them. Part of my intention is to model for you what good notes in a high-level mathematics course might look like. The notes will be posted in the e-classroom.

If you would like to read along in a book, the textbooks that are the closest to what we will be doing in ANA-I are Elementary analysis: the theory of calculus by Kenneth Ross, or Understanding analysis by Stephen Abbott. I highly recommend either of these two textbooks for additional reading or reference.
A free-to-download textbook that is not completely unsuitable is Basic analysis by Jiří Lebl.
The book Real mathematical analysis by Charles Pugh is somewhat harder than the level of this course, but is nicely written. Mathematics and bioinformatics students especially may come to appreciate this book greatly later in their mathematical careers.

Topics

We will begin by discussing what we mean by a real or complex number. We will start from the natural (counting) numbers, and use these to construct the integers, rational numbers, real, and complex numbers. The most difficult of these steps is that of the reals.

We'll go on to discuss sequences of real or complex numbers, and limits (plus related notions) of sequences. Series (infinite sums) are a natural next step, and we will discuss convergence or divergence of series.

Next, we'll discuss functions of a single real variable, including some of the basic topology of the real line. The definition of limits will be quite similar to that for sequences. We'll go on to look at continuous functions, and compactness of closed intervals. We'll finish with a brief discussion of some elementary functions.

Course format, exams and grading

Homeworks will be posted weekly to the e-classroom. Homework assignments, as measured by weekly quizzes, will comprise 20% of your grade. Quizzes will be given in tutorial sections. (So, the homeworks will not be graded, but problems closely related to homework problems will show up on a quiz, to be given during tutorial.) The remaining 80% of your grade will be determined by exams.

There will be 2 equally weighted midterm examinations (kolokviji). The first will be roughly halfway through the semester, and the second will be around the last week of the semester, at mutually convenient times (outside of lecture) to be determined. I highly recommend that you study for and take both midterms, even if you don't feel quite ready for them.

You may replace your midterm exam grades with your grade on the final examination (izpit). If you are highly successful on the midterm exams, you need not take the final. If you are successful on one midterm exam but not the other, you may make up the exam portion of the grade with 30% of the successful midterm and 50% of the final.

In equation form, your grade will be determined by

20%(homework) + max( 40%(midterm1) + 40%(midterm2),  30%(midterm1_or_2) + 50%(final),  80%(final) ).

In accordance with university policy, there will be at least 4 opportunities to take an instance of the final. However, you may only take the final once more after you have attained a passing grade.

Midterm and final exams will be in-person. Please show up at least 15-20 minutes before the start time, especially on midterms. You may not use a calculator or other computational aid, but you may bring one hand-written A4-sized sheet of paper with you to each exam.

Collaboration

I expect much of your learning to take place while working out the homework problems. You may collaborate with other students on the homeworks. Your solutions to quiz problems should be written in your own words. Solutions copied from another student or from the internet constitute academic dishonesty, which will not be tolerated.

Obviously, exams will be strictly your own work.