How to figure out the "idea behind" proofs in analysis? I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty figuring out the motivation for a lot of these or what the inequalities actually mean.
For example, I'm currently in Chapter 8 of Rudin, and have been trying to follow Theorem 8.2 for a while. I finally wrote out all of the steps in detail and I see how it all links up algebraically, but I'm not sure what any of it really means. Are the intermediate steps really just means to an end to get to an inequality you want?
Another example is that it took me about two quarters to realize that continuity intuitively could mean a function is almost constant in a small neighborhood. Insights like this help a lot, but I have difficulty following them.
How does one arrive at the "idea" of a proof?
Apologies if this is too vague of a question.
 A: The real problem which many students face in "real-analysis" is that they fail to comprehend the inequalities involved. It is important to leave the algebraic manipulation of $+, -, \times, /$ and really appreciate the power of $<, >$.
Consider this simple theorem: if $f$ is continuous at $a$ and $f(a) \neq 0$ then $f$ has same sign as that of $f(a)$ in a certain neighborhood of $a$. The proof is simple: since $f$ is continuous at $a$ we can ensure that all of its values are close to $f(a)$ and if we get too close to $f(a)$ the sign of these values will be same as that of $f(a)$.
This proof is as rigorous as a proof can be and it is not the product of symbol shunting but based on the understanding of inequalities. The basic fact of inequalities which we have used here is that if $A > 0$ then all numbers very near to $A$ are also positive (similar result if $A < 0$). More formally there is a neighborhood of $A$ which consists of positive numbers only. This fact is too obvious to state explicitly and yet the very simple theorem mentioned in last paragraph does not appear to be that obvious.
I think the reason behind the problem of "comprehending inequalities" is the extreme focus on algebraic manipulation in high school curriculum.
BTW not all theorems in real-analysis are as simple as the one I mentioned earlier. Some are complicated but the fundamental theorems are almost always that simple. Another point apart from inequalities is the concept of a real number. Any student who wishes to grasp the proofs in analysis must have a clear idea of
1) how real numbers are different from rational numbers and
2) how real numbers can be defined in terms of rational numbers (similar to the fact that how rationals are defined in terms of integers)
Both these facts are totally non-algebraic in nature and it requires some serious study of real numbers. And also the algebraic notion of symbol manipulation has to thrown away for a while to make room for appreciation of order relations.
Coming back to the proof of theorem 8.2 in Rudin, you need to learn one technicality of many analysis proofs. Most of the problems in analysis boil down to showing that $|A - B| < \epsilon$ where $\epsilon$ is a given positive number, $A$ and $B$ are quantities which depend on certain variables. In many cases the difference $A - B$ needs to be split into multiple parts and we show each part to be less than $\epsilon / n$ where $n$ is number of parts into which $A - B$ has been split. The strategy to show that each part is less than $\epsilon / n$ is different for each part and this requires a bit of study of some common proofs and a little thinking on part of the student.
