TITLE: Does notation ever become "easier"?
QUESTION [100 upvotes]: I'm in my first semester of college going for a math major and it's pretty great. I'm doing well, however, there seems to be huge gap between how difficult /complex an idea is and how convoluted it is presented. 
Let me make an example: In Analysis we discussed the Bolzano Weierstrass theorem and one of the lemmas showed that every sequence in $\mathbb{R}$ has a monotone subsequence. The idea behind the proof with the maximum spots ( speaking colloquially here ) is super simple and pretty elegant if you asked me, but I spent a significant amount of time trying to understand the notation of the professor until I went to this site to read a "proper explanation" of the proof, which had much simpler notation in it. 
Extracting the idea of the proof took me lots of time because of the strange notation, but once you understand what is going on, it is really easy. Most of the time spent studying lectures is about digging through the formalities. 
Do I just have to spend more time really going through all the formal details of a proof to become accustomed to that formality? Or do more advanced mathematicians also struggle to extract the ideas from the notation?
I'd assume there will come a point, where the idea itself is the most complex part, so I do not want to get stuck at the notation, when that happens.
( Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence if you are interested )

REPLY [12 votes]: I think I disagree with @littleO's comment that 

...one of the main difficulties in learning math is that often things are
  not explained very well...

I used to share this view. But actually I think that it is more accurate to realize that things really are just complicated and understanding them well only really ever happens in your own mind. And this takes a long time.  
When you are first trying to understand some math, a certain (probably oversimplified) way of looking at it might make most sense to you (e.g. just the "main idea" presented in a non-rigorous way). 
In order to master this piece of math, you might then read various different explanations, think about it in various different ways and talk to different people. As you do this, you are learning little extra facts, testing out and stretching your understanding etc. until eventually you understand this piece of math a lot better. And once you understand it well, the explanation that makes most sense to you will have changed. Then, miraculously

you finally find an explanation by someone who has grokked the idea

But in reality, the important point is that it is you who has finally "grokked" something. If you had been given this explanation in the beginning, you may not have found it so helpful because you may not have been ready to hear it.
To remark on your specific example of Bolzano-Weierstrass/existence of monotonic subsequences: The idea behind the proof is elegant and simple and maybe you really did understand it quickly. This is great!
BUT, (and I can't stress this enough):
the proof itself is not the same as the idea behind the proof; nor does it just consist of writing down the idea behind the proof
So it makes perfect sense that the proof may be fiddly and not straightforward to understand, whereas the main idea is simple. 
You asked 

Do I just have to spend more time really going through all the formal
  details of a proof to become accustomed to that formality?

One typically has to spend lots of time going through rigorous proofs in order to understand them on a detailed level. Once you build up experience and knowledge in a certain area of mathematics, this will become easier. (It will become easiest for proofs in that area, because a different area of math will have different notation and different conventions and so not all of this experience is transferable).

Or do more advanced mathematicians also struggle to extract the ideas
  from the notation?

It is absolutely true that more advanced mathematicians also have to spend a long time extracting ideas from proofs. But again the level of knowledge vs. the context does matter: An experienced mathematician who has been through many, many proofs in basic analysis will be able to skim such proofs for the main ideas. But when reading a new, technical research paper will still ultimately have to spend time going through the proofs carefully to slowly build up intuition.