TITLE: How to make my proofs and analysis more rigorous?
QUESTION [0 upvotes]: A similar question was asked here but is yet to be answered.
The problem I face is that when I try proving certain statements be it from geometry or algebra I get the general idea on how to approach it for instance using AM-GM Inequality but I am not able to set up questions in the form to properly apply the property. Similarly in geometry questions even though I know how to go about solving the problem like I figure out that a given set of triangles are congruent in the figure proves my result but I can't seem to find out how exactly to prove them.
As far as practice is concerned I try to solve almost all my study material problems myself and look up solutions only when no hope is left but still when a question which is different from the typical questions arrive I am unable to find a reasonable approach during the exams or even when I am practicing alone.
Any advise would be very helpful

REPLY [0 votes]: The short answer is that practice makes perfect.  A longer answer, that works for me, is: many textbooks and lecture notes present things the opposite way round to how they're developed.  Modern textbooks sets the goal out in the statement of the result (Theorem, Proposition, whatever) and then give a step-by-step process (with steps of varying sizes depending on the target audience) of how to get there.  But when you're faced with trying to establish a result, generally we do that by identifying steps that might reach that goal, and then refining each step until they're small enough for us to be certain that they are safe.  Older textbooks (from a hundred years ago or so) actually tended to present the steps first in a discursive format and reach a statement of the result at the end (and they didn't always write it as clearly as "Theorem:" either).  
So, perhaps an example instead of abstract wittering?  Let's take something on my desk at the moment: Given a Banach space $\ell_2^2$ show that its norm $\|\cdot\|_2$ is locally uniformly convex.
First things first: I write down the definition of locally uniformly convex so that I know what exactly I have to work with and what I'm trying to prove.  This means I have a sequence $\{x_n\}_{n\in {\mathbb N}} \subset \ell_2^2$ that converges in norm, so $\|x_n\|_2\rightarrow \|x\|_2 \in \ell^2_2$ as $n \rightarrow \infty$ and for which $\lim_n \|x_n+x\|_2 = 2$.  I need to show that $\lim_n \|x_n-x\|_2=0$.
Next thoughts: I know that not all Banach spaces have a locally uniformly convex norm, so that means that I cannot prove this using only generic properties of a norm.  That's a critical thought, because it tells me that I need to use the specific properties of the $\ell^2_2$ norm in order to prove this.  So at this point I write down the definition of the $\ell^2_2$ norm because I know I have to make use of that.
Next thoughts: convergence of sequences in norm is pretty standard in Banach space theory, so I should look at the additional condition and see what I can do with that.  I've obviously got a triangle inequality: $\|x_n+x\|_2 \leq \|x_n\|_2 + \|x\|_2$ and as soon as I write that down I can see that I can now apply the norm-convergence to get $\lim_n\|x_n + x\|_2 \leq 2\|x\|_2$
And finally, for this, I note that this is $\ell^2_2$ and so I have the parallelogram identity to help me out, and now I've found all the big ideas: I can see I can use the parallelogram rule to get $\|x_n-x\|_2$ in terms of $\|x_n+x\|_2$ and the convergence in norm to make all of the terms not involving $\|x_n-x\|_2$ disappear.  What's left is now filling in the details between each big step.
[By the way, I saw you updated your question on meta: Unfriendly Behaviour in Maths SE so it's probably worth me noting that this question is probably both too broad and too opinion-based to avoid closure, for which I'm sorry.]
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