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I'm still quite early in my Journey of Mathematics and as I have progressed I have found a constant wall; proofs. Every time I try and learn to do simple proofs, I struggle to find an actual strategy, and locating what I am actually trying to prove. I feel as if that when I construct a proof, I try and reuse exactly what the textbook has told me, but in the end I still get the proof wrong, because of not noticing a small detail or not managing to construct my OWN proof. I am wondering if there is a general strategy, or way to learn to just construct your own proofs without relying on outward sources.

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    $\begingroup$ Like most things in life, practice makes perfect. Start by trying to do lots of simple test questions and slowly try harder and harder ones. ideally try to work with more experienced colleagues who can also guide you. You will soon get the hang of things - just be patient my friend. $\endgroup$
    – Mufasa
    Oct 27 at 22:24
  • $\begingroup$ I always start doing first some easy examples, and then extreme cases - then when already have some insight of the problem (because always "something new" happen in the examples), I start to looking for the proof with a better "intuition" of where I have to aim. $\endgroup$
    – Joako
    Oct 27 at 22:25
  • $\begingroup$ Just read a lot of them and do a lot of them. I don’t know any better way. Perhaps start with something like point-set topology, which is a lot of definitions and proofs about definitions. You learn a particular set of skills with that subject. Building intuitions about the definitions, and then converting those intuitions into formal proofs. $\endgroup$ Oct 27 at 22:28
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    $\begingroup$ There are good books that will help you. The classic is Polya's "How to solve it". $\endgroup$
    – Rob Arthan
    Oct 27 at 23:05
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    $\begingroup$ One of my favorite books to recommend is Houston's How to Think Like a Mathematician. $\endgroup$ Oct 28 at 4:42
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Different take: Dont focus of proofs. Focus on solving problems. Then try to explain your solutions as clearly, concisely and rigorously as possible.

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    $\begingroup$ I agree, this is very helpful. Try to flesh out the answer in detail, not just the "general idea how to answer". I'm sure you will start noticing how to ensure the minor details are correct. $\endgroup$ Oct 27 at 23:07
  • $\begingroup$ I see, I think this would work for me, it just seems that the intuition of the worked solutions isn't something I would find without prompting. Details such as knowing that all multiplies of 6 are consecutive numbers, are things I just don't notice when trying to prove things are multiples of 6. What I am trying to say is that starting proofs, and knowing what to prove is not intuitive to me, and I would like to know how to work on that. $\endgroup$
    – Ammardian
    Oct 28 at 0:11
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First of all, to learn how to produce proofs one must learn how to read proofs. Say you're studying real analysis and you really struggle with $\epsilon-\delta$ proofs. This is a common occurrence. What makes you better at it? First, try to see a lot of examples. Each mathematical discipline has a certain flavor and so do the proofs of the results in that subject. So read a lot of proofs and go through many examples in the theory. That is the first step. Next, before you actually read a proof, try to think of what the person is trying to prove. Is it reasonable? Do you have an idea that might help you prove it? Read the proof. Is it what you expected? Did they use all the hypotheses? Where were they used? Keep asking yourself these questions and things will get easier. Lastly, try to revisit proofs yourself and try to prove more basic results on your own. Do a lot of exercises. This is the hardest part of studying mathematics in my opinion: gaining mathematical maturity.

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  • $\begingroup$ I try to do these things, but probably the thing is I often don't know where to start with my proof. Here is one I am doing right now, "If x⋹Q and y∉Q, prove x+y∉Q", For this I don't really know where to start, as I have seen examples of similar proofs, but don't know really where to start for this one specifically. It seems like a basic proof, but I don't really know what to do. $\endgroup$
    – Ammardian
    Oct 28 at 0:08
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    $\begingroup$ @Ammardian: Try to construct a proof by contradiction: If $x+y\in Q$, what does that imply about $x + y$? Can you say anything useful about $x + y - x$? If you later discover that the contradiction is unnecessary, you can then simplify the proof into a more direct version, but you need to have a valid proof to start with. $\endgroup$
    – Kevin
    Oct 28 at 6:48

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