dudeman12 said:
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 haven't had much experience with proofs outside of linear algebra and geometry, so in other words i don't 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?
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 \forall and \exists.
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.
