# Should I remember the proof of mathematical theorems(every step)?

The problem is, that when I am reading the proof of mathematical theorem(in my case - it is calculus), U understand the idea and every step of proof. But i can't prove the theorem individualy even if i understand all steps, that are written in book.

Should i concentrate myself on proofs of theorems or it is enough to understand the idea and process?

• It depends. What is your goal? It's one thing if you want to be an engineer and use the theorems (but don't really need to prove them on short notice), it's a different thing if you want to become a researcher in math, and it's an entirely different thing if you need an A or a 10 on a calculus exam so you can get into Medical School. – mathguy Apr 16 '16 at 13:43
• I want to work as financial engineer. And as i understood it is enough to understand the idea and the process of proof? – Daniel Yefimov Apr 16 '16 at 15:28

Example: in real analysis, do you remember how to prove that if $x_n, y_n$ are sequences which converge to $x, y$, then $x_ny_n$ converges to $xy$? You do so by splitting up the difference: $$|x_ny_n - xy| = |(x_ny_n - xy_n) + (xy_n - xy)| \leq |x_n - x| \cdot |y_n| + |y_n - y| \cdot |x|$$ And now you use the fact that since $y_n$ is a convergent sequence, it is bounded, and hence $|x_n - x| \cdot |y_n|$ goes to $0$ as $n$ goes to infinity.
I thought the argument of adding and subtracting $xy_n$ was elegant, but unintuitive when I first saw it. I didn't memorize that proof, but I did pay a lot of attention to that part. And, as a result, I am able to recollect the whole proof, and that technique of splitting up $x_ny_n - xy$ I have seen several times in other places in real analysis, in particular $L^p$ spaces.