at the moment, I have big troubles to solve exercices for my analysis homework. This is not a problem with an exercise in particular, it really is about how to solve things. I just started a bachelor's degree in physics and I have no mathematical maturity. I can easily use mathematics to solve physics problem but I have no idea what I am doing with "pure mathematics" (analysis).

I fail to understand the logical steps I have to make to prove anything.

I would like to know if there was some sort of methods/guidelines/anything really, I could follow for some cases that would help me see the steps I have to make to prove something and more importantly, how to know that my result is valid. How to know that I need to do a proof by contradiction or something else ?

I asked the same question to the teaching assistant and I am waiting for an answer but I thought that I could ask other people to have maybe more help.

How to get better at abstract reasoning for unknown solutions ? For example, I understand the proofs in class, I can do them again but I am totally unable to think like that myself. I really don't understand what my objectives are when I read an exercise.

Here is an example :

If I solve the inequality I have (x-b)(x+b) <= 0 but after that, I don't know what I need to do.

x must be equal to b or -b for the inequality to be equal to zero. But what about the inequality lower than 0? I can try to guess values of x to get a result lower than 0 like if x < b, then I get a product < 0.

So the possible values for x to to satisfy this inequality are b, -b and x < b. All those values satisfy the x^{2} <= b^{2}. Is that a satisfactory answer for the first part ?

After that, how do I answer the rest of the question about a and b ?

This sounds more like a homework problem but I didn't want my post to turn like this. My main problem (I am sure I have others) is not about a specific exercises but I have no process in my head that I can follow and/or apply to analysis problems to solve them.

I am more looking for generalization like if I have an inequality ab >= 0 to prove, I should try to prove ab < 0 and if I get a result which is impossible, I would have proven by contradiction that ab can only be greater or equal than 0. This is an example to help me visualize a process which will help me solve that kind of exercises;

You have to analyze these problems in steps: there are very few which can be solved by using this formula or that. This is the essence of analysis: to analyze the problem.

You want to find out what values of x make this relation, (x-b)(x+b) ≤ 0, true. You know that x = b or x = -b satisfy the equality portion. What happens if x > b? If x > -b?

Sometimes, it comes down to a process of exploring a finite number of possibilities.