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This is a very soft question.

Just a bit of background:
I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his exercises a bit difficult, and am always left astonished when a grad student tutor seems to arrive at the same conclusion I have in a fifth of the time. As such, it's prompted me to ask:

How much does one's mathematical ability improve year by year?

For example, how, in your personal experience, has your ability to write proofs and see through a problem increased over the years. Say, how was your first year of undergraduate proofs in analysis, to your second year, to your first year as a grad student. Naturally the growth will slow quite a bit eventually, but I suppose I'm curious to know whether this occurs right away, or 10,15,30 years off into someone's career.

It often feels as though, while doing a particularly difficult problem, that no matter how much you learn or practice, your ability would not allow you to see a solution. I want to know how much of this fear is myth and how much of it is truth.

I received an A in analysis I, but I often felt that if this, just undergraduate analysis, was difficult, then why bother to even try to do more difficult work.

Any comments appreciated, feel free to share personal experience, or to speak in a general sense.

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    $\begingroup$ A lot. ${}{}{}{}$ $\endgroup$ – GFauxPas Apr 13 '15 at 1:54
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    $\begingroup$ Don't worry, it improves significantly. The more you do the better you get and you soon come to realise that a lot of proofs just follow the same structures. $\endgroup$ – Rammus Apr 13 '15 at 1:56
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    $\begingroup$ It increases enormously. Not only does more mathematical maturity and experience proving theorems help, but I think you also get smarter as you become a little older. I'm in my early 20s, and I often revisit things I found difficult a few years ago, and usually find that they're much easier to approach now. At the risk of sounding trite, you are only 16, and if you're already studying real analysis now, you will be in an excellent position when you enter college (assuming you keep moving at at least your current pace). $\endgroup$ – Newb Apr 13 '15 at 2:37
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If it's alright, I'll just offer my own personal experience.

Like you, I also studied real analysis out of Rudin as a high school junior. I have since learned that Rudin's text -- while very efficient -- is notoriously concise, and that there are far more student-friendly texts out there.

Like you, I was often very frustrated with my inability to solve problems, create proofs, and understand the proofs given in the text. Though I ended up receiving A's in nearly all of my math classes, I sweated and toiled through each one. Very little came easily.

And like you, I clearly remember going to a second- or third-year graduate student's office hours, and sitting there dumbfounded as he easily breezed through problems that had taken me hours. I simplistically assumed that he was brilliant, that I was not, that things were simply going to be this way, and that I would never attain that level of fluency.

It's now 8 years later, and I've attained that fluency.


For me, there were two points at which my ability to prove things and "see through a problem" significantly improved. The first was when I completed the standard undergraduate sequence of courses (real/complex analysis, algebra, topology, etc.), about four years later. The second was roughly three years after that, when I had finished the graduate sequence of courses (and started reading papers).

I don't know why I had those mental growth spurts. Some part of me thinks that it has something to do with having a broader perspective, and seeing the bigger picture. But just as much of me thinks that it's about the sheer number of hours I'd committed to math, in various forms.


On a somewhat more personal note, I remember it used to bother me when professors and older students would insist that "seeing the big picture" would somehow, magically make problems clearer -- especially when my own on-the-ground experience felt counter to that. At the time, I felt a sharp division between technical mastery and conceptual understanding.

And indeed, I do think that there's a difference: technical ability and abstract conceptual ability can be different things. But lately (and only lately), I've found the two to be merging for me. On the one hand, I'm seeing just how big the big picture really is, and have been using it to solve problems quicker. At the same time, a friend's exam-time recommendation that I "focus on the methods" has led me to understand certain concepts better.

In short, many things which seemed magic and completely, utterly out of reach eight years, seven years, ..., and even as recently as three years ago, no longer seem to be so now.

Hope this helps.

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You need not be discouraged. Rudin often uses proofs that are elegant and concise, rather than easy-to-follow. I understand you desire to "see through" a problem, as you mentioned, but if you can at least follow what Rudin's proof is doing, then you are strengthening your mind for challenging proofs. In time, you will begin to see patterns in all the proofs you have worked through, and naturally, as you see new problems, you will begin to refer back to a proof you have seen before. You seem to be doing things just right, with a strong start. Be confident in yourself that your abilities will improve at a significant rate, if you stay the course.

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