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.
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.
Well, that is exactly what I did, and I got my first problem set back, and I got a 79! I never got a score that low on a homework before, and I honestly was so confused as to how I could have messed up that much when I was so confident in what I had turned in.
Introduction to Real Analysis
So, after class, I met with my professor, and we went through it, and I saw now that I really needed more assistance and time to learn not only technique, but I had to learn the new notations, the difference between evaluating and proving, and just make my math more mature as they say. With this, I immediately scratched my plan of doing the second problem set and all the rest by myself, and I found myself going to my first ever office hours because I regretfully shied away from going to any during my first semester.
I thought it would be intimidating working with a professor on the homework, and I thought I would come off as dumb or not knowledgeable enough to even be part of the class if I went to office hours. I was so wrong.
The biggest takeaway I got from my second semester was to take advantage of office hours literally as much as possible.