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.