Struggling with Analysis I? Get Advice Here!

[muzik87]

Hello!

This is my first time here on this forum! (Hopefully I am posting at the right place). Anyways, I am writing this to have some kind of advice. I am taking Analysis I in university right now, and it seems to give me a little bit of a rough time. I noticed that I will go to class, understand what the prof does in class and everything, but when comes the time to do exercises or problems (involving proofs most of the time), I seem to have no clue where to start, or I simply am doing it wrong or it's totally wrong. But I DO understand the class to some point, and I usually follow what the prof is saying.

So any advice on how to get past that "wall" involving the way I could do problems and have a general idea where to start?

Thanks!

 

[AlphaNumeric2]

Though I can't directly answer your question, I can tell you that I've been exactly where you are. Lectures make sense, proofs given are logical but when it comes to constructing your own solutions, haven't got a clue.

Stick with it. Even if it doesn't fall into place immediately or even 6 months down the line, things slowly bubble away at the back of your head and it'll piece itself together over time. I've had moments of clarify about things 2 or 3 years after some lecture courses where it just 'clicks'. Though I'm nowhere near as good at analysis as my pure maths friends, the "Oh dear god no!" element of the subject has gone.

A good book for analysis is Burkill, Amazon usually has some cheap copies available.

 

[symbolipoint]

Would a particular course be an unstated pre-requisite for that Analysis 1 course? Would a "Strategies of Proof" course give necessary subject matter strength to more effectively work the problems and proofs for an Analysis 1 course?

 

[muzik87]

Would a particular course be an unstated pre-requisite for that Analysis 1 course? Would a "Strategies of Proof" course give necessary subject matter strength to more effectively work the problems and proofs for an Analysis 1 course?

In fact, no... well not at the university I am at. I did briefly some proofs before, but really from a logical point of view (ie: P => Q format). But profs show you proofs in class sometimes, but they more than often think it's trivial stuff to show (as they probably think like you that we did take a class that shows us the rigorous way to do the proofs).

I did hear from other students taking the class this year that another class sort of helps when coming to proofs. I'm actually registered for that class NEXT semester (as for my program it was a second year class, only offered in the second semester), so hopefully it will help a little. But now, got to go with the flow and work harder! Hopefully all will fall in order so I can pass the class!

AlphaNumeric2: Thanks for your reply! I'm glad I'm not the only one feeling the same! I will try and check that book you recommended.

 

[Gib Z]

I've had a similar feeling, and am in fact having it right now, though to a smaller extent than before. I find that when I read the content before the questions, I follow fine and it all seems good to me, but when it comes to some of the questions I just have a large block. Sometimes its just a mental block, other times it seems the question requires some sort of prerequisite knowledge that I don't have or just expects me to notice some amazing way of expressing the problem, It can be quite annoying but I've found its actually very good practice! For instance I am doing Volume 1 of Calc by Courant at the moment, though I already know most of the content in the book. I find the questions and his new methods are very interesting, and that's the reason I'm doing it. Of course with these new methods, I very soon run into problems I can't do, which is good =]

Short Version: Don't worry, be patient, time will do its work.

 

[eastside00_99]

Most analysis proofs, at least at that level, start with a definition that is contained in the conditional part of the proposition. Other times, it is just an inclusion or exclusion argument or showing something is less than or greater than. Then there are the times where you have to let epsilon be greater than zero. That is probably where most of the trouble is and epsilon-delta arguments are not immediate for anybody. But, there will be a day maybe a year from now when it sinks in on a very deep level, and you sill scream "Ahha, give me a problem, I can do it". For me, it wasn't until a year after my first analysis course that I really felt like I prove just about any theorem we covered. Another thing to keep in mind is that NOT every math major will fall in love with analysis. Some will dislike it and love algebra. Doing algebra as opposed to analysis takes a little bit of different mind set. So, since is basically your first course in doing proofs don't let it discourage you from exploring other proof based math courses if you start to really dislike the material. I may be somewhat weird because I took my first abstract algebra course and analysis course the same semester and loved them both. I regretted studying analysis because I wasn't studying algebra and vise verse. But, I have to say algebra was more like second nature whereas analysis was difficult to get used to.

 

[SiddharthM]

analysis is that way for most people. if you stick with it, you will eventually gain proficiency. Remember one more thing - this is probably the first time you are expected to understand and perform non-trivial rigorous mathematics, so you are also learning how to learn this kind of stuff - and it is never easy.

eastside: I'm doing sum algebra 4 the first time now after having done about 4 semesters of analysis courses (i know, very strange) and it is freaggin AWESOME. It definitely isn't nower near as difficult as my first analysis class, but again - that's because at the time I didn't know how to learn "real" mathematics. Mathematical maturity is truly a phenomenon, very much like proficiency in a language.

cheerio

 

[HallsofIvy]

Most important rule: Learn the exact statement of definition! It is not enough to have a "general" idea of what something means. Proofs use the exact words of definitions.

 

[eastside00_99]

SiddharthM, that's an admirable way of doing it. Four semester of analysis is quite a bit and probably makes algebra seem pretty easy. If you know algebra, then linear algebra is pretty easy, and if you know linear algebra analysis and functional analysis is muche easier. So, the tools you learn in algebra will help with your study of analysis which you probably realize. anyway best of luck.

 

[Edgardo]

Hello muzik87,

1) If you need more practice in writing math proofs,
take a look at this thread:
How to write math proofs

So any advice on how to get past that "wall" involving the way I could do problems and have a general idea where to start?

2) To get past the "wall" I recommend this summary of George Polya's "How to Solve it".

 

[muzik87]

Wow! Thanks a lot people for the incredible advice you're giving me here. I really see a way for me improving at this without thinking it's going to be a VERY big obstacle, but rather a something that will develop over time if I put the effort in.
Those links you gave me (Edgardo) sure will help, and all the recommendation from everyone else sure helps!

Thanks a lot everyone!

 

[mathwonk]

do you ever read over the lectures afterwards to be sure you understand every step in the proofs? do you try writing out the proofs from class to see if you actualkly remember them? do you try specializing the theorems to see what corollaries they have? or do you just sit there and blissfully elt it fly over your head, telling yourself all the while, this sounds easy enough?