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I'm studying math for a while now and still, even if I understand the question, on my own I'll probably mess up with a very informal "proof" that is mostly verbal. Even if I write down the definitions and have them in front of me, I still am not able to nail a rigorous proof.

What do you recommend me to do in order to improve my proof writing skills ?

Note: this is for first year math at the university.

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You should aim your proof at your reader. Formality and rigor are fine, as long as they don't hinder understanding. Just formulas are useless, explain what is going on.

Get some classics on your chosen area (there are several lists of texts renowned for readability floating around), and study the style of the proofs. Try to imitate them. Check out texts on mathematical writing, and on writing proofs.

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