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Preparing for an Analysis Class

  1. Mar 3, 2012 #1
    In a semester or two i am going to take an analysis class that uses "principles of mathematical analysis" as its book and i am terrified since it seems like a really difficult book. i havent had much experience with proofs outside of linear algebra and geometry, so in other words i dont have much. i have been trying to through "elementary analysis: the theory of calculus" and i was going good until i got to these pseudo epsilon delta proofs that defined the limit for a sequence and i just got destroyed by every single problem and even after seeing the solutions i couldn't really apply what i saw to other questions. i feel like i am going to be crushed by the analysis course since i can't get very far in this book.

    so how would you guys recommend just preparing for a course like analysis since i can't get through an "elementary" analysis book without being destroyed?
  2. jcsd
  3. Mar 3, 2012 #2
    You have plenty of time to prepare. If you are worried about your calculus/proofs background, the only way to get better is to practice. I have not used the book you mention, but I know Spivak's "Calculus" is good prep for Rudin. Do lots of the exercises. If you are struggling with one, don't give up right away - sometimes they take a few hours to figure out. You can also get help here on PF.
  4. Mar 3, 2012 #3
    I loved that course when I took it. I have some suggestions. This came out long, my apologies in advance.

    * Relax! This is very fun course. It's not about math as in pushing symbols around till you get the right answer. It's about math as in: Finally understanding how modern mathematics handles infinite and infinitesimal processes. Since the time of the ancients, mathematicians used infinite and infinitesimal reasoning, even though it made no logical sense. In the 17th century, Isaac Newton systematized calculus, but STILL never really understood how to deal with the underlying logical absurdities of the subject. It wasn't till the 18th century and even well into the 19th century that mathematicians figured out how to put the whole business on a sound logical basis.

    So this course is first and foremost the story of one of humanity's most profound intellectual achievement: creating a logically satisfying account of how to handle the infinite and infinitesimal processes of math, using finite reasoning that can be checked step by step for correctness. We've only nailed this down a little over a century ago. In math that's not a very long time. No wonder it's hard! But it's exciting too.

    This course is an intellectual adventure. It's really cool.

    * I disagree with others who say you should review or study calculus. Sure, you need to have gotten through a year or two of calculus, pull down the exponent and subtract 1 and all that. But frankly you don't really need it. I could never do those integration problems. But I was a whiz at epsilon-delta proofs in real analysis. It's all about understanding what they're talking about. You spend a lot of time learning the concept, and then the symbology is easy.

    * Assuming you mean Principles of Mathematical Analysis by Walter Rudin, rest assured this is one of the best and most beautiful math books ever written. It's the right book for the job.

    * This is going to be an exciting adventure. That said, you must allocate a substantial amount of time for it. If you were taking this class and doing nothing else in your life, that would be ideal. Work back from there. Don't take too many other classes. Don't take another difficult class. Set aside as much time as you can. Whatever you can manage.

    * As you noted, one aspect of this class that trips up newcomers is the idea of a logical proof. But there are a lot of tricks, and you'll pick them up. One trick is to go overboard with clarity. Write down what you're being asked to prove. Write down specifically what technical fact(s) you are required to prove. Then go step by step until you've shown exactly what you needed to.

    If you have an "if and only if" then you really have TWO proofs, a forward and backward direction. This kind of thing ... all these little techniques that make up doing proofs.

    You might surprise yourself and find that you really enjoy this style of math. If you're the kind of person who likes to be "right" about things, this will help :-)

    * If there's one subject you should brush up on, it's basic set theory. Not only is set theory the foundation of everything you'll be doing in class; but set theory is full of proofs. Set theory proofs are like the warmups for real analysis proofs.

    So you should know what they mean by set, subset, proper subset, element. You should know union, intersection, power set. The DeMorgan laws. Functions, relations, Cartesian product. Do lots of proofs. You should do all the set theory proofs people ask here in the homework section. Show that this expression is equal to that expression. How many elements are in the power set of the power set. All those kinds of things. I think facility with basic set theory is the single most important prerequisite.

    And you should become familiar with writing and manipulating expressions involving the universal and existential quantifiers [itex]\forall[/itex] and [itex]\exists[/itex].

    Personally I would not spend much or even any time reviewing calculus. It's just not relevant. And don't worry that you can't jump into epsilon-delta proofs right now. Real analysis is not about the epsilons and deltas. It's about the concepts underlying the epsilons and deltas.

    Well thanks for reading if you got this far. If hope something I said helped to put this class into context for you.

    Also if you are fortunate and have a gifted teacher, this will help alot. I strongly recommend forming a small study group with a few other students. This is a subject that really benefits from the process of exchanging ideas with others learning the same material.
    Last edited: Mar 3, 2012
  5. Mar 4, 2012 #4
    The book you mentioned by Ross seems like a good book to prepare you for Rudin. I would say you must work through this book before doing Rudin. Rudin is much more difficult and assumes you know the stuff in Ross, so it is highly recommended you do Ross first. I would probably go so far as to say that you will not understand Rudin at all unless you go through a book like Ross. On your first reading you can leave out all the starred sections.

    And I don't know what you mean by "pseudo epsilon delta proofs", though I haven't read the book in question. I suspect you mean proper delta-epsilon proofs, because that books looks to be a proper, rigorous book. If you can't master the concept it just means you have to work harder. If you have trouble you can always ask a question here on PF.

    If possible try to take the course in 2 semesters rather than 1, so you can really work through Ross. Or if you must take it next semester, I would say a bare minimum is the first 3 chapters of Ross for preparation, minus the starred sections, although that may not be enough.

    Good luck.
  6. Mar 4, 2012 #5
    The key to epsilon-delta proofs is practice and time. It's not abnormal to have troubles with them, I didn't understand them either when I first saw them.

    Reading Spivak's calculus is a good idea. The book deals with epsilon-delta proofs in a clear manner. If you work through the relevant sections of the book, then it'll be easier for you.
    One could say that Spivak isn't really a calculus book, but more of a pre-analysis book.

    I don't like Rudin's book at all. You shouldn't only look at Rudin. Look at other books as well. Ross is also a good book. Another good book is "Understanding Analysis" by Abbott.

    Finally, ask a lot of questions on the material!!! PF is a resource you should use.
  7. Mar 4, 2012 #6
    i say pseudo epsilon delta proofs because there is a part of the book where you have to find the limit of a sequence and it defines it only in terms of epsilon, but the process of proving the limit of a sequence is similar to the epsilon delta stuff.

    thanx guys, i guess i just have to tough it out and work through this book
  8. Mar 5, 2012 #7
    The book is correct. The test for the limit of a sequence only involves epsilon. It seems to me you really are having a hard time with this stuff. Maybe it's best to go back to the first proofs involving delta and epsilon (or just epsilon), and start again.

    Good luck.
  9. Mar 5, 2012 #8
    When I took Analysis, I studied from Rudin and Apostol. I think for a starter Rudin on its own might be a tough so having a complementary book will help. As others noted above, it also depends how good the teacher is and whether you will have people to work with.

    ++ for spivak if you want preanalysis. There are more qualified people in this forum to give a better advice but here is my 2 cents based on my experience taking the class.

    --Don't expect to understand everything first time. There are so many definitions and theorems that might rather seem arbitrary but as the course develops and you take other maths classes, you see why things are defined that way and why certain theorems are important.

    -- Don't be a proof machine. Don't just think of every analysis problem in terms of delta epsilons. See what the problem implies visually/geometrically. It might be hard to always do this but most proofs and techinical details are so much easier to absorb if you get the jest of it.

    --When new definitions come about, think concerete examples. Say you learn about a compact space. Ask yourself what are examples of compact sets? What are some examples of spaces that are not compact? Why is compactness such a big deal?

    Good Luck
  10. Mar 5, 2012 #9
    I just want to reiterate micro's point about Rudin because I mostly agree with his view. I I won't say I don't like Rudin at all...Rudin is great book, but you have to realize that it is written in the most concise language possible. It does not EXPLAIN anything. Rudin's value lies in its exercises and as a reference that contains everything you're supposed to know from introductory real analysis.

    I think a better book for insight is Elementary Classical Analysis by Marsden and Hoffman. They provide pictures explaining concepts and take more time to explain things. Another good book is by Bartle (can't recall exact same, something including the word 'analysis').
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