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My Experience with Real Analysis by Jack-William B. '21

epsilons, delta, and the technique

June 10, 2018

Last semester, I took 18.100A (Real Analysis) with Professor Choi. Throughout my entire time in high school, I used to complain to my math teacher about how I wanted more of the proof and verification for all these theorems. Well, in the most MIT fashion, I can defiently say that I got more than I asked for. However, looking back, I could not be happier that I decided to take the course that honestly was overbearing at times, and I even have decided to change my primary major from physics to mathematics due to that class in particular.

Going into 18.100, I saw the material was based on one-variable calculus. 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. 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. I can, without shame, say that I literally attended every single 18.100A office hours from the second pset to the final exam, and I don’t think I’ve ever taken a class and gotten so much out of it before. I did start going to my 8.03, Physics III, office hours as well, but that is a whole other story with an amazing TA.

Anyways, 18.100 was the first math class I have ever taken where I didn’t feel like I could just learn the material nonchalantly if that makes sense. I was used to being able to easily grasp things without much practice, but this class was a whole monster by itself. I have about two notebooks worth of me going through practice problems using the various techniques we used. By the end, when I was reviewing for the final, it was honestly humbling to look back at all my failures of somewhat proofs compared to how much they improved by the end.

The experience for me was one I will never forget because not only did it show me how much I love the challenge of that type of mathematics, but it also showed me that not all classes are created equal. While I would spend maybe 15 hours on that class a week, I would spend barely 3 hours on other classes in my schedule. I saw that it was much more rewarding to take a course that was seriously challenging than to just sign up for a course that I know I could easily get a good grade in. More than any other class before, I learned how to study for exams, I learned how to manage my time, and I learned how to reach out for help.

Luckily, I had an amazing professor, and I really hope I get to take another class with him in the future because I really like his teaching style. When I came into the course, my intuition for solving problems has been completely turned around because of the time during office hours where he carefully went over the ways we should apply different techniques and theorems to different forms of problems. He would sometimes say rememver the technique, and I could not help but picture spongebob screaming the technique as I tried to focus on my homework and exams! Going into the final, I felt a lot better than going into the midterm, and I managed to get an A in the class not because I was born with some amazing ability to solve problems and cite Cauchy, Bolzano, and Weierstrass theorems out of thin air, but because I worked harder than I ever have to get the best understanding of the material.

Next semester, I have a good sense of what classes I am going to take, and I am most excited to take 18.701 (Algebra I). 18.100 turned out to be both the class I was most worried about and also the class that I loved the most. I can’t wait to see what the future holds in my other math classes.