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

I'll assume that you have a reasonably strong high school mathematics background.

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.

If you would like to read along in a book, the textbooks that are the closest to what we will be doing in AN-I are *Elementary analysis: the theory of calculus* by Kenneth Ross, or *Understanding analysis* by Stephen Abbott.

A good 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 what we'll be doing, but is beautifully written. You might appreciate this book greatly later in your mathematical careers.

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. Limits, continuity, and compactness of closed intervals. If time remains, we'll examine additional topics.

Homework assignments will comprise 20% of your grade. 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 to be determined.

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. 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.)

Thus, your grade will be determined by

20%(homework) + max( 40%(midterm1) + 40%(midterm2), 80%(final) )

You may bring one hand-written A4-sized sheet of paper with you to each exam.

I expect much of your learning to take place in working out the homework problems. While you may collaborate with other students on the homework, I expect you to have thought hard about the problems on your own first. If you do collaborate substantially, then you should so indicate on your homework paper.

Your write-up should in any case represent your own solutions, written in your own words. Copying the solutions from another student or from an internet source is a form of academic dishonesty, and will not be tolerated.

Obviously, exams will be strictly your own work.

*Last modified February 26, 2018*