# Best way self study real analysis?

1. May 13, 2013

### Government$In October of this year i will start with math major and i decided to prepare myself in spear time. In my faculty there is no such thing as Calculus but rather you go strait to the analysis and you pick up calculus along. (There is singe variable calculus in high school). In first two years subjects that have most ECTS (Grading system in EU) points are Anlysis 1A,1B,2A and 2B where each lasts one semester i.e. in total two years. And in third year i have Analysis 3A,3B and all of these courses are proof based. Topics for Analysis 1A/1B is single variable calculus and series and for Analysis 2A/2B multivariable calculus and things like multiple remain integral and Fourier series. And for Analysis 3A/3B topics are measure theory and functional analysis. To prepare my self for Anlysis 1/2 i decided to go through following: 1. How to prove it by daniel j velleman 2. Understanding Analysis by Stephen Abbott 3. Baby Rudin 1. Do you suggest me to add something or to use some different material? 2. And what is the best way to learn RA? The way i see it it is best for me to read definition and theorem, read and understand proof and then to try to prove same theorem myself and then move on to the exercises? 2. May 13, 2013 ### dx For analysis 1/2 I recommend Spivak's books "Calculus" and "Calculus on Manifolds." I too began studying analysis as an undergraduate with Rudin, but I only started to enjoy it and see its beauty once I found Spivak. 3. May 13, 2013 ### mathwonk I think Abbott will be more useful than Rudin, or a useful supplement. They are at opposite ends of the spectrum of user friendliness. One should be aware however that after the standard Riemann integral, Abbott treats the "generalized Riemann integral" which is a non standard topic, whereas Rudin discusses the more usual Lebesgue integral. Last edited: May 13, 2013 4. May 13, 2013 ### Government$

So you mean to use Spivak's textbooks instead of Abbott or to just add Spivak to the mix?

I know that Rudin doesn't have great rep for user friendliness. I thought using Abbott first as an intro and then to study Rudin.

5. May 13, 2013

### dx

Just add it to the mix. Its always good to see the same thing explained by different people. (I've never read Abbott by the way.)

6. May 13, 2013

### mak52810

Yoou may find the video lectures of Prof. Francis Su from Harvey Mudd College useful. Hes on YouTube.