Any advice to help visualizing/understanding theorems in Real Analysis? I am currently taking a real analysis course and sometimes I can't really get the complete meaning/details of certain theorems.
For example, one of the statements in the Heine-Borel Theorem saying that:

Every open cover for $K \in  \mathbb{R} $ has a finite subcover. 

I have difficulty visualizing what's going on here...Most of the textbooks only prove the theorem in words,however,after reading those lines of proof I could only have a vague idea of it.
Would you give me some advice please? Or if there are any good web resources/reference books to help visualizing theorems in Real Analysis?
 A: Well, it depends on the book. The statement of this theorem seems like it's assuming some other types of analysis/ general topology (esp. since many of these theorems are generalized in topology.)
However, that's the trouble with specific math courses, they tend to actually assume some mathematical maturity from other courses you have studied. So, check out an equivalent statement that doesn't involve jargon (but is more difficult to state):
Let $C \subseteq \mathbb{R}$ be a closed bounded interval. Let $I$ be a nonempty set and $\{A_{i}\}_{i \in I}$ be a family of open intervals in $\mathbb{R}$ [so that they are just a bunch of $(x,y)$ being indexed by $i \in I$.) suppose that $C \subseteq \bigcup_{i \in I} A_{i}$. Then there are $n \in \mathbb{N}$ and $\{i_{1},..,i_{n}\} \in I$ such that $C \subseteq \bigcup_{k=1}^{n} A_{i_{k}}$.
This pretty much means that if you have some $[x,y]$ in $\mathbb{R}$, and it is expressed as a subset of any bunch of open, then you can express it as the union of a finite number of open intervals.
So in an easy case:  $[0,1]=C$. Let $\{(-\infty,2),(2,\infty)\}$ be a family. Obviously, $C$ is a subset of a finite union of elements in this family of open sets, namely, the both of them. 
However, the real point of the problem, is that if a closed bounded interval can be covered by a bunch of open intervals, you can find a finite covering. This seems like a technical result but actually has some interesting implications.
To see why it has to be a closed interval: consider $C=(0,1)$. Consider the family of open sets: $A=\{(1/n,1)|n \in \mathbb{N}\}$. This covers $C=(0,1)$ [you can check this.] But there is no finite subcover, because if you pick any one of them for a large $n \in \mathbb{N}$ as "the last one" you can always find a smaller $\epsilon>0$.
In terms of your broader question, it just takes practice. I read from an introductory analysis book at first that spends an unusual amount of time on the least upper bound property. It's called: The rea Numbers And Real Analysis by Ethan Bloch. There is also a great series on youtube given by Harvey Mudd that spends an excellent amount of time on real-number construction, which is helpful (especially with preliminary theorems such as Heine-Borel.) Generate some examples and have fun with it!
