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Don't get me wrong, I don't like to see hard to answer, ambiguous, subjective questions brought up either, but I could really use some advice.

My problem is the following: If I can't solve problems (usually involves proof-writing) that can be dealt with using Maths I was taught, then does this mean that I'm lacking enough talent and ingenuity (I most probably am) to solve them? Or Does it imply that I need to be more familiar with the 'proof models' and 'tricks' that surely exist in the field, but don't come up as theorems and lemmas?

After finishing high school, and becoming an undergraduate (I'm not doing a career in Maths), I am now faced with serious difficulties (Preparatory Classes if you need to know), back then we were doing relatively easy problems that only required a good grasp of the ideas presented in the class, these could be mastered if you did enough of them, didn't require much thinking or creativity, and were divided into many simple questions ...

Now I'm expected to solve this kind of problems:

Let $P$ be a polynomial of degree $n$, $n \ge 1$. Prove that the equation $P(x)=e^x$ has a finite number of solutions.

With nothing but Rolle's theorem. I know it's doable, but I couldn't do it, the proof I saw was 'beyond my level', it's like a playground I can't reach, but only hope to understand.

Eventually, and in only two years from now, I would be asked to prove this kind of things on my exams:

Let $\{a_n\}$ be a bounded sequence of positive real numbers. Show that the following statements are equivalent:
$(i) \text{ } \frac{1}{n} \sum_{k=0}^n a_n \to 0$
$(ii) \text{ There exists a subset A of } \mathbb N, \text{ with null density (meaning } \frac{A\cap [\![1,n-1]\!]}{n}\to 0 \text{), such that } \lim_{\begin{smallmatrix}{n\to \infty}\\ {n\not \in A} \end{smallmatrix}} a_n = 0.$

It's becoming more and more plausible to me that to be able to solve these type of dilemmas would require diving into a lot of problems, that one's level would be directly proportional to the number, and difficulty of problems they've dealt with, assuming everyone revises their lessons (consider it given).

But is this Maths? The naive part of me still wants to believe that all this is some kind of game where you set up the axioms and everything else falls into place with pure deduction, no need to know how all those past mathematicians approached problems and wrote their proofs... I think I was wrong.
But then what should I do? Do I just give up trying to 'come up' with solutions myself? (Nothing remotely brilliant has ever existed on my notebooks unless the professor wrote it), and start memorizing all the tricks I'm missing? Will that be machine work, or would I actually learn something of intellectual value?
My brain is a pure deduction machine, if I saw something like this problem before, I solve it, otherwise, I keep looking at it and filling pages of failed attempts and speculation with no results.

I managed to write a terribly long rant about my incompetence as a Maths student, all this given that I'm not even pursuing a career in it, I apologize if you felt you had to read all this and I hope you can advise me, I'm not ready to give up just yet nor am I looking for some magical method, I simply want to know the best way to move forward without turning myself into a bookshelf and Maths into history.

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  • $\begingroup$ Out of curiosity, what subject are you studying? I'm wondering what subject, other than math, would require you to prove that theorem on your exams. (Especially since I think it's false as written.) $\endgroup$ – saulspatz Nov 3 '18 at 16:54
  • $\begingroup$ Exactly, I'm not studying any actual subject, I'm only preparing for exams to enter universities. I'm not studying maths for anything particular. And I need to be that good in problem-solving. $\endgroup$ – FuzzyPixelz Nov 3 '18 at 17:01
  • $\begingroup$ False? Which one? $\endgroup$ – FuzzyPixelz Nov 3 '18 at 17:12
  • $\begingroup$ The one about the bounded sequence. That doesn't look right to me. Suppose $a_n = 0$ if $n$ is even and $1$ if $n$ is odd. Then the average partial sum goes to $1/2.$ But $A = \{2^n | n \in \mathbb{n} \}$ is of null density. $\endgroup$ – saulspatz Nov 3 '18 at 17:20
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First of all, it's apparent that you're not enjoying your Mathematics! Undergraduate Mathematics is just the beginning of the beautiful aspect of what really Mathematics is! Until +2, whatever maths one encounters is nothing but some basic definitions and sheer computations. That's the amazing part of Mathematics, the more you study higher Mathematics, the lesser the computational troubles and more the Conceptual proofs and Understandings.

Yes, of course Mathematics is just a language based on certain axioms (as it appears) but there are a lot more to it than just that. There is a certain amount of aesthetic feeling regarding doing something so much abstract! Motivation should generate from inside.

And regarding difficulty in writing or coming up with proofs, more you face difficulty and you still keep trying it on, the more you'll be familiar with the setup, more fluent with ideas! Definitely not all proofs have the same idea and simply forget about remembering all the tricks etc. , One can be assured that there are not enough tricks to exhaust all problems and that is again a beautiful aspect of the subject.

Instead of taking pressure on doing problems, simply try to enjoy the beauty of the subject. Think of exploring it! That would definitely help!

Cheers and Best Of Luck!

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  • $\begingroup$ As much as I want to enjoy how beautiful Maths is, this doesn't help make me any better when I deal with problems. $\endgroup$ – FuzzyPixelz Nov 3 '18 at 17:12
  • $\begingroup$ Classes préparatoires are very intensive and can put a lot of pressure on the students. Sadly, mathematics there is a ruthless selection tool and not a beautiful thing to explore. $\endgroup$ – mercio Nov 3 '18 at 19:21
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In higher level math, there's often a necessity to transcend what you've learned to synthesize something new. This is true in any subject, but because students are mostly only taught to do mathematics by rote, they've had no practice doing this. This is, in my opinion, a failure of secondary school education more than anything else.

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  • $\begingroup$ Could you please elaborate on the first sentence? It's a little vague to me. Also would you have any advice to students on how to "transcend what you've learned to synthesize something new"? Is it just doing more hard problems and not focusing only on learning theory? $\endgroup$ – Ovi Nov 20 '18 at 5:06
  • $\begingroup$ @Ovi For me, the key was spending hours upon hours secluded in my room working on hard problems. There was a course I took where I might spend 4 hours on a single problem. Because of my patience I got $100\%$ on all the homework. $\endgroup$ – Matt Samuel Nov 20 '18 at 11:40

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