Best way to learn pure mathematics I'm currently enrolled in an introductory real analysis course.
While I enjoy the beauty of pure mathematics, there are 2 problems I face.
1) Constructing proofs
2) Attempting to understand a proof without seeking help despite hours attempting to understand.
While I do in general understand many of the proof, I find it difficult/ impossible to construct the proof. Is this a matter of a lack of practice?
Should I be asking help in understanding the proof whenever I am facing a brick wall? Should I be expecting pure mathematics to be time-consuming?
 A: *

*and 2. are hardly unique to your case. Literally no mathematician has ever avoided these two "problems" (I prefer to call them "challenges") at any stage of their careers.



"While I do in general understand many of the proof, I find it difficult/ impossible to construct the proof. Is this a matter of a lack of practice?"

Yes. It's hard for everyone, especially at first. Keep at it and you'll be astonished at how your mind adapts. "Mathematics is not a spectator sport."

"Should I be asking help in understanding the proof whenever I am facing a brick wall?"

In the early stages of your education, only after trying your absolute hardest to understand the proof, consulting other references if need be, should you perhaps ask for a hint about a proof or a solution. Be honest to yourself - only ask for help when you have actually exhausted all other options. You only hurt yourself by asking for help prematurely.

"Should I be expecting pure mathematics to be time-consuming?"

Yes. It's hard, and that makes it worth doing. Individual results in mathematics (pure or otherwise) are culminations of weeks to months to years to decades of thoughts and ideas. Centuries if you include your predecessors that added to the foundations of these ideas. If you are studying mathematics full-time and your problems do not take you very much time to solve, chances are you should be looking at harder problems.
A: If you are interested in pure mathematics, and would like to give a try, I recommend Linear Algebra (it is on its 4th edition now, i guess) written by Friedberg and other two authors. In fact, this was my first book studying in this field, and I found it profitable to me.
Most of students encounter the two problems you mentioned, so don't be that depressed. One of the reason (and I believe, the reason of most time) of being unable to construct a proof is not understanding all the definitions, which let one be confuse with what should be proved. The book starts from the definition of vector space and also offers enough explanations with intuition. Moreover, I believe the problem sets it contain helps a lot too, especially the true-or-false problems. Try to give every details in the beginning to support your proofs, and you will improve.
The second book I learnt from is Baby Rudin, which is not easy at first glance, but I found it profitable to copy his proofs. You will learn from it a lot :)
A: It is practice sir. Math is one of those things where, if you have a calm, relaxed mind, things will just come if you give it time.  So take your hours to understand your proofs, you will find which way you learn best.
