How to develop the patience in mathematics?
When you read the books like Rudin, Conway, etc., it takes a lot of time for one problem; sometimes $10$ hours each.
I usually devote one problem for $30$ minutes. If I can't solve the problem within $30$ minutes, then I post the problem on this site for a solution.
Actually, I have no patience and can't wait for long hours for thinking about the problem.
How can I improve myself and develop the patience?
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Whenever I need to study something new, I study with friends. When you do that, as everyone is thinking in the problem, you solve problems faster, and it is easier to keep yourself concentrated.
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Reading solutions step by step might be a good habit to increase the patience. By that I mean, when you struggle with a question for $30$ minutes, if you have access to a solution to that problem, you don't have to read the whole solution. For example, first you can check the first idea and after understanding it, you can simply keep struggling through that idea for another $30$ minutes maybe. If you are still not able to solve it, you can go to a further step in the solution and take it from there for another $30$ minutes, not reading the whole solution.
However, there are few things to add here. There are some problems which are hard to understand but easy to solve. For those kind of problems, the solution might be a one line proof for instance so you may ask "how can I go through the solution step by step if the solution is too short?". In these kind of cases, I would suggest trying to have a better understanding on question. This could be achieved by reading more about the topic or turning back to the points where you could not grasp the intuition. Note that this kind of "struggling" also develop some kind of patience.
Finally, there might be some questions for which you don't have an access to solution. In this case, I suggest trying to find similar problems to the one you have. Sometimes, it is more satisfying (and even harder) to adapt an existed solution of another problem to your problem, than solving the problem itself.
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Firstly, if you are asked to prove a statement, think a counterexample even though the question say only prove (not disprove or give counterexample.) Write your all examples and check them which properties are satisfied and not satisfied. This is my method and I am very happy with that. By the way, please be patient in general. Learning something always take time. The point reached without climbing never be top.
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$4$ years ago I started my journey. Initially , used just can't sit for more than half an hour with simple real analysis stuff. I found that reading multiple books has given me insight how basically theory develops and I look at similar problem. For better insight. Even if I have solved to get a better understanding(but I do this in semester breaks else my grades suffer a lot). I just push myself to do assignment by setting alarm for let's say 10 or 20 or 50 minutes. For me initially I thought that doing assignment without proper understanding is not good but my here in Indian pattern my professor said I am a tourist guide who will show you various places but to know your places and how long you have to stay depends on number of days you are with me.
So, you should choose wisely. If you are stuck ask help. Some problems might take a month or so just to show existence but after that it's cake walk.
And I asked my professor same question that you just asked he says it's part of planning.
Your planning improves with practice. He says even I plan how to cover course and make my lecture interactive over years still I was not able to do so($15$ years of experience) . Keep practice - know your constraints - optimize output.
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While developing patience is one thing, the counter-intuitive answer is that if you are taking too long to solve problems is that you are not solving enough problems. Mathematical problem solving is something that needs to be trained, both in general, but also in every subject.
If I, as someone without much training, were to run a marathon right now, I would probably take 10 hours to do so. But if I give up after 30 minutes, no one would tell me that I need more patience. A much more reasonable suggestion would be to first try to get into shape for a much shorter distance.
The same is true for textbook problems. There are of course outliers, but there is no reason for a normal textbook problem to need anywhere close to 10 hours of dedicated time to solve. If you are stuck at one problem, then you should try an easier related problem and come back later. This might come from the same, or a different textbook or can even be something you come up with on your own by modifying a given problem (e.g. dropping a condition and looking for a counterexample to something you're meant to prove).
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Briefly skim through the provided answer. Don't memorise it. Come back at a later date and see if you can now solve the problem. If you can, you have internalised the method. If you can't, you still don't understand it fully.
Rinse and repeat.
Skimming through the work of others is a useful research strategy to develop. Without it you will get bogged down reading even one paper that includes original work.
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Terence Tao statement taken from https://terrytao.wordpress.com/
The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.
Michael_Atiyah statement taken from Advice to a YoungMathematician
''Be patient and persistent. When thinking about a problem, perhaps the most useful device you can employ is to bear the problem in mind all the time: it worked for Newton, and it has worked for many a mortal as well. Give yourself time, especially when attacking major problems; promise yourself that you will spend a certain amount of time on a big problem without expecting much, and after that take stock and decide what to do next. Give your approach a chance to work, but do not be so wrapped up in it that you miss other ways of attacking the problem. Be mentally agile: as Paul Erd˝os put it, keep your brain open'' You have to be practice patience, because by practice everything can be acheived .Practice make a man perfect
Abel prize winner lenart carleson statement taken from his interview
If you want to solve problems, as in my case, the most important property is to be very, very stubborn.Stubbornness is important; you don’t want to give up. But as I said before, you have to know when to give up also. If you want to succeed you have to be very persistent. And I think it’s a drive not to be beaten by stupid problems.
It's all part of the character building nature of the subject. But I find that it helps to have multiple questions on the go; once I can no longer stand one problem, I just put it aside, work on another problem, then return to it or yet another once I'm impatient with that one.
Take it in incremental steps: do five minutes more, then ten, twenty; an hour, two hours; an afternoon; a day, a week, a month, a year!
Note that some textbooks deliberately put in open problems as exercises.
Also, if you're planning on doing a research degree at all, you've got to learn which questions are fruitful and which to leave unanswered, bearing in mind that some questions can take decades to answer.