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I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to.

In introductory undergraduate classes in Analysis, usually, Rudin is assigned which has a pretty straightforward way of introducing topics. The author provides the definitions, then concludable theorems along with their proofs and at the end exercises.

My main concern is with the proof for the theorems. When I am going through the proof for the theorems, what should my aim be?

Should I be making sure that I understand each step, that's to say, that I will be able to understand and explain the proof provided that I am given the proof?

OR

Should I try to read the proof again and again so as to be able to rebuild the proof from memory and intuition alone (and of course understand the proof in the process)?

Possible Constraint I would like the reader to keep in mind:

I am aware the optimal strategy is to try to approach the proof myself first, along with coming up with counterexamples for the conditions in the theorems, however, given the fast pace of these courses and the fact that an undergraduate is not just taking the analysis course, really minimizes the time a person can devote to such sort of analysis.

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    $\begingroup$ How to best make use of the available time is going to depend on the emphasis of the specific course you are taking. Some places are focused on applications, so are more interested in you being able to do the computations, whereas others are more focused on the theory. $\endgroup$ – Jessica B Mar 6 at 20:59
  • $\begingroup$ Of possible interest is my answer to Lesson plan to self-teach real analysis to student with comp-sci background. $\endgroup$ – Dave L Renfro Mar 8 at 17:49
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There are two separate things you want to take away from a proof, and they should be the focus of your study.

The first important takeaway from a proof is the driving idea (or ideas) that make the proof work. You should generally be able to express these ideas using as much ordinary language as possible. As an example, I will consider the theorem that the uniform limit $f$ of a sequence of continuous functions $$f_n : D \subseteq \mathbb{R} \rightarrow \mathbb{R}$$ is continuous. (This can actually be considered as a special case of a more general theorem, but that's no matter here.)

The driving idea behind the proof is to use the triangle inequality to bound $$|f(x) - f(x_0)|\, \text{ by } \,|f(x) - f_N(x)| + |f_N(x) - f_N(x_0)| + |f_N(x_0) - f(x_0)|$$ for all $x$ in a suitable disk of radius $\delta$ centered at $x_0$ using a suitably large choice of $N$.

The three terms in the latter summand can be made less than $\epsilon/3$ because $f_n$ converges to $f$ uniformly, $f_N$ is continuous, and $f_N(x_0)$ converges point-wise to $f(x_0)$, respectively. Note that this part of understanding a proof is creative and requires understanding the subject in a greater context.

The second important takeaway from a proof is how to turn the driving idea of the proof into a rigorous argument. In the example above, this is where you manipulate $\epsilon$'s and $\delta$'s and $N$'s appropriately, carefully considering the order of quantifiers and what depends on what. Note that this part of understanding a proof is technical and can be improved upon with constant practice.

When you are first learning proofs, the second part can be very challenging. The more experience you get, the more you will come to realize that the first part is the generally harder part for discovering a new theorem. The second part can largely be automated, but the first part requires a genuine insight. (Constructing new proofs is actually a bit more complex than that. Generally, one comes with lots of potential strategies for a proof, then analyzes whether they can be turned into rigorous proofs. Often, there is a gap in the argument, and then one must go back to the creative stage and figure out a way to close the gap or a new approach to the proof.)

Of your two options, the second one is the one that will help you to learn how to approach proofs successfully. Focus on understanding the big picture and then on how to turn those ideas into rigorous arguments. Don't memorize proofs line-by-line. It's important that your goal should be to understand the proof, not to merely be able to recite it.

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    $\begingroup$ The driving idea behind the proof is to use the triangle inequality --- For the OP I mention that, for those already sufficiently conversant with basic techniques, simply saying it's an epsilon over three proof is usually enough for them to know what's involved, although getting to this point is what the actual question is and where the difficulties lie. One thing that might help right now is to classify absolute value inequality proofs you've seen as "epsilon over two" or "epsilon over three" (or more, but those are rare). $\endgroup$ – Dave L Renfro Mar 9 at 15:51
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The latter of your two choices. You don't know it if you can't recreate it. Just reading stuff and saying "yeah I get it" is the lazy way and you don't really learn from it. This was true in technique based math courses as well or chem or physics or languages. (Bio/history/lit/psych/etc. can be read a bit more passively with large gains of knowledge. However, even here some "drill" will make them better.)

I think it's OK not to bang your head against the wall too long to independently derive every proof while reading (not needing the text at all). However, at least a short attempt at first deriving the proof will make you more invested in the text once you do read it.

After seeing the proof, practice rederiving it. It will approximate the learning from first deriving. Hopefully the exercises as well will build some skills into your brain. This is especially because it is near impossible to just regurgitate everything as a memorized script. Thus some of it you will internalize in terms of building the ability to rederive it.

When you are practicing, try to read the proof and then put the paper away and see if you can redo it yourself with a blank paper. If needed, look at the book, but only for the smallest quickest hint and the go back and finish the proof. If you had to look at the text (at all), force yourself to do the problem right over with no peeking. This is really a similar study skill to doing technique based problems as well. You need to do this in order to grind some grooves into your head, to internalize the knowledge.

If you are struggling with Rudin, try other easier texts like Cummings or Abbot.

https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-Textbook/dp/1724510126/ref=sr_1_3?keywords=real+analysis&qid=1551898925&s=books&sr=1-3

https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116/ref=sr_1_4?keywords=real+analysis&qid=1551898925&s=books&sr=1-4

I feel you on having other courses and needing to be time efficient. But some level of moderate struggle is needed to drive learning. An hour of reading a text like a novel (or for "recognition") is far inferior to an hour of working problems. So it's actually time inefficient to be too weak and passive in the learning process. As said, this was the same in your technique courses.

P.s.s. I have never taken a real analysis course and I'm not a teacher. (Caveats.)

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