### Course Description

Thus, the real numbers comprise integers, rational numbers, and irrational numbers. You are probably familiar with the association of real numbers with points on a line, the real number line. Here's an important question: are there points on the line without real number names?

That is, are there any holes in the line? One final issue: each real number represents a finite quantity, but the set of all real numbers is infinite. How can we use the real number system to describe define?

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My course begins with a review of the axioms of the real number system, induction, and cardinality. We then meet the important Completeness Theorem, proceeding, as time allows, to limits, sequences, series, continuity, differentiation, and integration. This is a course in real analysis for those who have already met the basic concepts of sequences and continuity on the real line.

## Real analysis

Here we generalize these concepts to Euclidean spaces and to more general metric and normed spaces. These more general spaces are introduced at the start and are emphasized throughout the course.

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• A comprehensive pack of lecture notes will be provided. The following may prove useful:. Now, I knew the focus of the class was to teach you how to write proofs and not necessarily focus on the actual calculus material as being difficult, but I still went into the class a tad bit overconfident and not as concerned that the class would be as difficult as I thought it would be.

The first week went fine, and we got assigned the first of 7 biweekly problem sets, and I completed it all by myself. I thought, okay, easy enough, I just do this and that, cite a theorem, and turn it in.