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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 :

Let b ∈ R be a fixed real number, solve the inequality x^{2}≤ b^{2}. Give a necessary and sufficient condition on a and b so that a^{2}≤ b^{2}.

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

Thank you.