The best way to prepare for real analysis

[Benzoate]

okay, i already take an intro to proofs class. did average in it. I'm taken real analysis in the spring, but as a degree audit course. the semester following my spring semester , I am take real analysis for a grade.whats the best way to prepare for real analysis?

 

[mattmns]

What book is your class going to use? One suggestion would be to start going through the book now. Also, you said you did average in your proof writing class, so my suggestion would be to go through all that stuff again and make sure you understand it well (proof writing practice will definitely help you in your Real Analysis class).

I think the best way to prepare would be just practice doing proofs, post questions here if you have any. More importantly, be prepared to spend as much time as you need to understand everything. Do not fall behind. If you have any questions, post them here, and/or ask your professor at office hours. If you put in the effort (time), you should be able to pass. I went to my professor's office nearly every week when I took the first and second semesters of Real Analysis, and I got an A+ in both. It was not easy, and I spent a great amount of time on both classes (especially the second), but it was worth it.

 

[Benzoate]

What book is your class going to use? One suggestion would be to start going through the book now. Also, you said you did average in your proof writing class, so my suggestion would be to go through all that stuff again and make sure you understand it well (proof writing practice will definitely help you in your Real Analysis class).

I think the best way to prepare would be just practice doing proofs, post questions here if you have any. More importantly, be prepared to spend as much time as you need to understand everything. Do not fall behind. If you have any questions, post them here, and/or ask your professor at office hours. If you put in the effort (time), you should be able to pass. I went to my professor's office nearly every week when I took the first and second semesters of Real Analysis, and I got an A+ in both. It was not easy, and I spent a great amount of time on both classes (especially the second), but it was worth it.

how many hours did you spend writing proofs. I went to my professor officer everyweek but showed no great improvement in writing proofs.

 

[mattmns]

I am not sure I understand what you are asking. You really just need to keep practicing doing proofs, that is the only way to get better. My first proof class (abstract algebra) was brutal, I considered changing my major because of it, but I stuck with it, and during the break I read that book I recommended (doing all the exercises, and making sure I understood everything), then next semester I took my second proof class (linear algebra), and I did really well. The following year I took Real Analysis 1 and 2, and did well in those classes. There I spent maybe 10-25 hours a week doing homework and reading for each class. I think for proofs you just need experience (the more you do, the easier they become). Just watching your professor do a proof probably won't help your skills too much, but you should think about their thought process (and even ask them why they went some route).

Also keep in mind that proof writing can be ugly. When you see a proof in a book, or in lecture, you are seeing the final result that has been tweaked to have a nice logical flow. When you are actually trying to figure out how to prove something it is much different; you will try many different paths before ultimately coming up with all the steps. So don't think that every step should just come to you naturally and logically, try things. Another suggestion is to question everything. Whenever you see a theorem, try to prove that the theorem is wrong (come up with a counterexample). Of course you should not be successful in proving that the theorem is wrong, but in the process you should come up with some intuition for why the theorem is correct. Another thing to do is when you come to a theorem in a book, cover the proof, and try to do it yourself. If you get stuck, then read the first line and try to finish it yourself.

 

[proton]

real analysis shouldn't be too bad if you aced proof-based linear algebra, right? (as in your midterm scores were twice the mean)

 

[morphism]

Real analysis shouldn't be bad at all, provided you have a good book, a half-decent instructor, and enough motivation to do the required work.

 

[rbj]

i took Real Analysis (which was in the math department) as an EE student about 3 decades ago. the text was by H.L.Royden. the reason why i took it was that i wanted to get into a good formal mindset so i could take 2 semesters of Metric, Banach, Hilbert Spaces and Functional Analysis (this stuff is very useful for stuff we do for Communications Systems) and one semester of Real Analysis was sort of a pre-req.

i liked it and did okay (i think i got to B), but, even to this day, i still have trouble with an issue i brought up recently in another thread.

after having taken undergraduate courses in Diff Eq (incl. Laplace Transforms), Linear System Theory (now a lot of schools are calling it "Signals and Systems"), Probability and Random Processes, Control Systems, and Communications Systems, we Electrical Engineers had a working and practical understanding of the Dirac Delta "function" that got turned on its head by the formal mathematical understanding of it (that i first was exposed to in this Real Analysis course).

when we learned (in Real Analysis) that given two functions f(x) and g(x) that are equal "almost everywhere", that their integrals must also be equal, that was a direct contradiction of what many EE texts said about the Dirac delta (what we EEs called the "unit impulse function") which is equal to zero everywhere except x=0, yet the unit impulse function had an integral equal to 1 (not zero). this is also where i first heard that the Dirac delta function "isn't a function", but something else.

this difference in pedagogy is something that has bothered me ever since, including the times that i have taught EE classes. pedagogically, it just is easier (in an EE class) to skip this fine distinction between whatever the Dirac delta really is (a "distribution") and a regular function. but then we run afoul of what the formal mathematics is. nonetheless, nothing ever broke.

it seems whatever the forum is, mathematicians officially disapprove of the electrical engineering concept of the Dirac delta. but it really is a difference of concept between two disciplines, it's not just my own personal misunderstanding of it.

 

[morphism]

Not this again! rbj, matt addressed your 'issues' in the other thread - which you also interjected. If you're still not satisfied, start a new thread in the appropriate forum.

 

[mathwonk]

you might try reading a good calculus book like spivak. its basic analysis but written more clearly for freshmen.

 

[JasonRox]

I find Analysis so freaking hard! I find it hard to imagine there exists a good textbook for me.

I'm visiting the Basic Analysis and Complex Analysis courses next term (I took them already) just to try and understand it better. The profs. that are assigned for these courses actually teach. If any of you remember my Complex Analysis frustration, you know why I'm going back. It's a different prof. this time. THANK GOD!

 

[Nolen Ryba]

I love seeing threads like these, because they remind me that even within something as specialized as mathematics we all have our differences. I'm a grad student and honestly I have a hell of a time with algebra, topology, and number theory, but analysis comes easily. Maybe it is just the layers of abstraction that cause difficulty for me, but anyways, I would recommend Ross's "Elementary Analysis" for the basics of real analysis. Rudin's "Principles of Mathematical Analysis" is a little more advanced but still "undergraduate," i.e. it ain't screwin the pooch but it also isn't truly hard core. If you want to "cut to the chase" you might as well pick up either the Stein series or the Rudin series and dive in :)

As far as rbj goes, man I hear ya- I may be educated in math/physics but I've learned to use the delta function in applications. I'd guess that I'll eventually end up in that line of work, even though I'm currently a math grad student. I've recently learned the theories of measure/Lp spaces and distributions and while they are indeed clarifying and complete, the requirement of extreme details tends to make things a little less clear intuitively. The comment you make, however, is irrelevant because the the theory of Lp spaces does not take into account entities like the delta distribution. In the context of distributions this does make sense.

 

[mgiddy911]

I just finished an Analysis I class using Arthur Mattuck's Introduction to Analysis (had my final on Wednesday)
I would say there's not much you can do right now to prepare as you are already familiar with proofs. The only thing I can think of has already been mentioned, that being review your calculus from a rigorous book like Spivak, Courant or Apostol.

Be familiar with proofs using Epsilon and Delta, as they are incredibly important
You could try reviewing sequences and series as that will be what the beginning of the class covers.

The class I took was Analysis I, it was one semester and meant to be continued the next semester for Analysis II (however I'm taking part II at a later semester)
Thus my class covered sequences, series, a little bit of Set Theory (Cantor's Theorem and such), and the beginnings of Calculus, ie. Functions of one variable, limits of functions, Bolzano's Theorem, IVT, MVT and Rolle's Theorem, Extrema
We did not get to Integration

But based on the above you can see some topics to review. You may not realize just yet how in depth the class will go we spent half a semester or more on sequences ans series, there is an incredible amount of knowledge there.
Mainly I would just say when the class begins, do all the required homework even the stuff not to be turned in. Get a feel for the style of proofs used in the theorems, and practice proving the theorems so that you can reproduce them on tests
hope that helps out