Studying Real Analysis

Question

I'm studying Real Analysis at university and I'm finding it difficult. I am understanding the concepts, but very gradually. We are going through material on Continuity and Limits, Differentiability, Power Series, Integration and require prerequisites such as Logic and Sequences which I need to brush up on greatly. We have just completed lectures on Differentiability. I'm finding the lecture notes hard to grasp and I'm contemplating reading through the recommended text 'How to think about Analysis' by Lara Alcock, from scratch. I would really like a solid understanding of real analysis and how to approach this subject better. Is it a good idea to read through this type of book before the lectures get further into the subject? Is there any book on Real Analysis that you could recommend. Thank you.

Answer 1

I asked someone how to do well, and they told me that they just make sure they understand the proofs from class/have them memorized for the test. Everyone that is successful is doing that. So I guess just start preparing sooner for the tests, like everyday, just sit down and go over the material from that day's lecture. Real Analysis essentially is all about real numbers, so try to think about everything in terms of closed/open intervals. Arbitrary metric spaces can be employed when the proofs are harder in the reals. It's sometimes called the theory of calculus, but the more real analysis you take, the more you realize that it's really it's own thing.

People tend to find abstract algebra to be a bit easier/less hand-wavy at my university. It's alright if you find it hard, because it is hard. I struggled a lot in that class and wanted to switch out of math altogether, not because I don't understand the concepts, but I have a hard time coming up with proofs. The trick is just to practice a lot with it. Go to office hours for the homework if you need to, I certainly did, and a lot of the kids that were doing better in that class lived in the guy's office and had already taken topology.

There's also video lectures on You Tube that I find really accessible. Best of luck!

Answer 2

I cannot agree enough that brushing up on your logic is invaluable. Consider the following definition of convergence of a sequence $(a_n)$ to a limit $a$.

$$ (a_n) \text{ converges to } a \iff \forall \epsilon \in \mathbb{R}_+,\exists N \in \mathbb{N}, \forall n \in \mathbb{N}, n \ge N \implies |a_n - a| < \epsilon $$

Being able to write down statements like that in symbols is invaluable to writing good analysis proofs. Being able to take negations of statements is equally important, as is familiarizing yourself with proof by contradiction.

My next advice is to try and solve lots of simple problems before you try harder ones that you're being assigned. For example, if you're learning about limits, try proving that $1/n$ converges to $0$ and that $1/n$ does not converge to $1$. That too hard, try showing the constant sequence $(1,1,1,1,\ldots)$ converges to $1$ and doesn't converge to $0$. If you cannot do the basics, you cannot do more complicated things.

As you slowly increase the difficulty level from easy problems to harder problems, notice where things become more challenging. This can give valuable insights in what your current struggles are over.

I've never read the book you're learning out of but it seems good. "Understanding Analysis" by Abbott is another good book pitched at a similar level.

I'm an undergraduate learning analysis myself. In my experience, there's really no substitute for doing exactly what you say you want to do in your question: really go back to basics and make sure you master each concept before moving ahead.

Hope that helps!

Answer 3

That depends on what's slowing you down. If you're having trouble developing an intuition for the material, try visualizing what various theorems are saying, or drawing illustrative examples on a sheet of paper. Once you have your foundational intuition, the material will be more readily absorbed. After a while, the need to rely on visualization will diminish, as the language of real analysis becomes more "native" to you.