Preparation for Real Analysis

[Ankit1010]

I'm taking undergrad Real Analysis I at college next semester, and I want some advice on how I can start preparing for the course over summer since I need to do well on it. I'd appreciate references to good books/online materials and any advice you might have about how to do well in the course.

Here's the course description:

The Real Number System: Field and order axioms, sups and infs, completeness, integers and rational numbers. Real Sequences: Limits, cluster points, limsup and liminf, subsequences, monotonic sequences, Cauchy's criterion, Bolzano-Weierstrass Theorem. Topology of the Real Line: Open sets, closed sets, density, compactness, Heine-Borel Theorem. Continuity: attainment of extrema, Intermediate Value Theorem, uniform continuity. Differentiation: Chain Rule, local extrema, Mean-Value Theorems, L'Hospital's Rule, Taylor's Theorem. Riemann Integration: Partitions, upper and lower integrals, sufficient conditions for integrability, Fundamental Theorem of Calculus. Sequences of Functions: Pointwise convergence, uniform convergence, interchanging the order of limits.

[Ben-oni]

I'm taking undergrad Real Analysis I at college next semester, and I want some advice on how I can start preparing for the course over summer since I need to do well on it. I'd appreciate references to good books/online materials and any advice you might have about how to do well in the course.

Here's the course description:

The Real Number System: Field and order axioms, sups and infs, completeness, integers and rational numbers. Real Sequences: Limits, cluster points, limsup and liminf, subsequences, monotonic sequences, Cauchy's criterion, Bolzano-Weierstrass Theorem. Topology of the Real Line: Open sets, closed sets, density, compactness, Heine-Borel Theorem. Continuity: attainment of extrema, Intermediate Value Theorem, uniform continuity. Differentiation: Chain Rule, local extrema, Mean-Value Theorems, L'Hospital's Rule, Taylor's Theorem. Riemann Integration: Partitions, upper and lower integrals, sufficient conditions for integrability, Fundamental Theorem of Calculus. Sequences of Functions: Pointwise convergence, uniform convergence, interchanging the order of limits.

Practice drawing lower case epsilons and deltas.[/snark]

Practice proofs. Look up any of the terms listed that your not familiar with. That should do you good for now.

[Qaanol]

All right, Real Analysis is a proof-based course. If you have not had one of those before, it will be quite different from your other math classes. No longer will the professor tell you how to solve problems and then ask you to solve them. Now you will have to prove that the techniques to solve those problems are actually valid. You should be familiar with proof by induction, proof by contradiction, and proof by contraposition.

The course will start out covering thing that you’ve known since elementary school, but have probably never studied rigorously before. For example, in the section about field axioms, you might be asked to formally prove that 0·1=0. You might have to rigorously define what it means for a real number to be “positive”, and prove that the positive numbers are closed under addition, multiplication, and division, but not subtraction.

From the list of topics you provide, it seems likely your course will not make you prove that such a thing as the real numbers actually exists. Instead it will implicitly take the stance, “If a complete ordered Archimedean field exists, these are the properties it must have.”

Most of the course will stem from a few major ideas. One of these is the least upper bound property, meaning every bounded set of real numbers has a least upper bound in the reals. Another is trichotomy, meaning for any two real numbers x and y, exactly one of “x=y”, “x<y”, and “x>y” is true.

There will be a lot of topics dealing with limits from the ε-δ definition. This will probably be couched in the language of neighborhoods and balls, and likely will constitute your first introduction to the study of metric spaces. The course will start slowly, then progress quickly, until at the end you will be rigorously proving calculus theorems that you might not have seen before.

As far as preparation goes, it will help if you are able to read and write mathematical proofs. You should be able to follow formal logical reasoning, and preferably able to produce your own. There’s not much in the way of “math” background that you’ll need. The course is mostly self-contained, and will build things up from basic arithmetic all the way to calculus, proving almost everything along the way.

[Ben-oni]

As far as preparation goes, it will help if you are able to read and write mathematical proofs. You should be able to follow formal logical reasoning, and preferably able to produce your own. There’s not much in the way of “math” background that you’ll need. The course is mostly self-contained, and will build things up from basic arithmetic all the way to calculus, proving almost everything along the way.

And it really is a ton of fun. If you like that sort of thing, anyways.

[Ankit1010]

Practice proofs. Look up any of the terms listed that your not familiar with. That should do you good for now.

...
As far as preparation goes, it will help if you are able to read and write mathematical proofs. You should be able to follow formal logical reasoning, and preferably able to produce your own. There’s not much in the way of “math” background that you’ll need. The course is mostly self-contained, and will build things up from basic arithmetic all the way to calculus, proving almost everything along the way.

I've actually done several proof-based courses before and am very comfortable with writing and understanding formal proofs. Let me clarify what I meant in the question - I want to know what textbooks/problem sets/video lectures/resources I can use to start learning the material that will be taught DURING the class, not the general areas I should cover for background knowledge. The fact is that my GPA that needs a lot of work, and next semester promises to be tough so I'm trying to effectively teach myself everything we do in class over the summer to get a head-start.

[nxcho]

Understanding Analysis by Stephen Abbot is a very good book for self studies and the toc is suspiciously close to your course description.
A good textbook in combination with wikipedia and http://math.stackexchange.com will probably do the trick.

[Qaanol]

Ask the prof what textbook the class will be using.

[Ankit1010]

Alright, thanks for the advice! Understanding Analysis looks like a great book, I'll definitely pick up a copy, and I'll find out which text we'll be using for the class soon too. It will likely be either Principles of Mathematical Analysis by Rudin or Understanding Analysis. I love stack exchange, and that coupled with this forum for more serious difficulties should be enough to resolve any problems.

[fishfry]

I'm taking undergrad Real Analysis I at college next semester, and I want some advice on how I can start preparing for the course over summer since I need to do well on it.

I had a wonderful academic experience in that course. There were two factors I've identified.

1) I had a fabulous teacher. You have some control over that if there's more than one section and you can find reviews. This is difficult material ... and for someone who wants to do advanced math or physics, it's absolutely essential to nail this course. So get the best teacher you can.

And along these lines ... buddy up to the TA's. The TA's are grad students who still remember what it's like to not understand this material ... so they can be incredibly helpful. Join a study group. Go to TA office hours. Be friendly. Hang around with the other students. It really helps to grapple with this material with other people.

2) I took it during summer school and took nothing else. In fact I was taking an upper division computer science class and just dropped it. I did nothing but real analysis. And it really made a difference. This is a very labor-intensive course. You just have to do epsilon proofs till they come out your ears. Because the course involves concepts that are deep; and techniques that are precise. You really have to put some time into this class.

That would be my advice. Sign up with a good prof; hang out with the TA'S and other students; and clear the decks in the rest of your life so that you can spend all your time thinking about real analysis.