(typo: title should be Feeling, not Feelings. Whoops) Hi there, I'm a second-year student at a highly ranked private liberal arts school and I am pursuing a BA in Math. Once I graduate, I want to pursue a PhD in either Math or CS. I obviously don't know exactly what I want to focus on, but Combinatorics was awesome and I have a fondness for Automata/Formal Language Theory in CS (and curiosity of subjects like Cryptanalysis and Information Theory). However. I'm facing some serious conceptual difficulties in Real Analysis. For some reason, I just have a really hard time even applying basic techniques (epsilon-delta proofs, etc.). I was also quite bad at Calculus. I have a 3.0 Math GPA and about a 3.4 overall GPA. Compared to peers at my school, a 3.0 isn't actually that bad, but it looks terrible on my transcript. My math grades have been strictly improving since my first semester, but I don't think I can maintain that this semester. I guess in short I'm just having an identity crisis. I really, really love math, and I'm quite good at *some* kinds of it, but unfortunately not all of it. I feel like I'm pushing myself into a little corner where if I don't get into a graduate program, I'm screwed--although I love computer science, I don't enjoy coding and although I love math, I don't like the kind of math I would do as an actuary/other applied math job. I feel awkwardly trapped between these two disciplines, and not very well-rounded in either. Do you guys have any advice or reassurance? Thanks.
The trick to epsilon-delta proofs is that you want to show that if one quantity is constrained, then another related quantity is constrained. If you can conceptually get the concept, the symbology is much easier to work out. Perhaps you can post some specific examples you're having trouble with in the homework section. FWIW I was bad at calculus but did well at real analysis. You don't need to be able to do related rate problems anymore until you have to teach them :-) But I think real analysis is an important class if you want to go to grad school, even in discrete math. In any event in computer science they care alot about bounding the growth of functions, and that's all about limits. So in the end you are going to have to grok limits. There's no way around it.
Yeah--I had mentioned to a friend that one difficulty I have had in Analysis is simply TRANSLATING the question into a concepetual idea I could work with. I found that in combo, when I didn't even know where to start, I could re-write the question in *plain english* and it would generally make more sense. I've found that translating these questions into english is MUCH harder and would take pages. I think part of my problem is just grappling with all the different definitions and things that I have trouble keeping all in my head at once. Sure. The problem that almost drove me to tears last night was a proof of the "Contraction Mapping Theorem". (4.3.9 part b in Abbott's "Understanding Analysis") The problem was: Let f be a function defined on all R and assume there is a constant c s.t. 0 < c < 1 and |f(x) - f(y)| <= c|x-y| for all x,y in R. Pick some point y_1 in R and construct the sequence (y_1, f(y_1), f(f(y_1)), ...). In general, if y_{n+1} = f(y_n), show that the resulting sequence (y_n) is a Cauchy sequence. I had NO idea how to even START. I actually didn't end up solving this problem. I more or less left it blank. Good to know, haha. Really, I think a lot of my problem is just not being comfortable working with symbols and not words. My brain is really wired for english and language, not symbols and "math". That's good to know. I was kind of under the impression that this wasn't particularly related to what I liked, which I think made me a little bitter about having to take it. I understand it's importance, I just hate it! :P I appreciate your use of "grok".
Don't worry buddy, EVERYONE feels that way when they start up with analysis. Wanna know why? Because there's nothing 'natural' about it, you really have no intuition on the subject at all when you start with it which is why you're having trouble translating the problems into concepts you can deal with or (if you're anything like me) having trouble translating your concepts into the language of analysis. There's no way about it, you just need to stick at it until one day it finally 'clicks'. You may perhaps want to check out several different books on analysis (specifically introductions), go back to the very basics. Learn how to and understand how to do all the little proofs at the beginnings and just work your way up. Good luck! :D Ps. Also, it's normal for your confidence to go as tan(t) in maths. I think everyone that does maths must have thought 'I'm so stupid, I suck at maths' at LEAST once a month!
I don't mean to be discouraging, but that isn't true. I got over 100% in my first real analysis class, I think with an extra credit question on each each test. It was really very easy for me. It was in grad school that I started struggling.