Any advice on getting better with proofs?

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Brian Moser
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When asked to prove something, I'm usually clueless. Even given the necessary axioms to use, I never know what to do. However, once I've seen a proof I understand what they've done. This doesn't mean to say that reading lots of proofs improves my ability to do them myself: all I can do is memorise them and hope that only the exact same proofs are asked for in future.

Any advice on what would help my ability to do proofs?


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Noble.
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It goes without saying that proofs are something most people struggle with initially, but it's something you get used to with time and experience. Later on, struggling with proofs is a good sign that you're not fully understanding/remembering the core material and if you find you're merely memorising proofs you're doing something wrong.

A good thing to do to help make a proof memorable (or even shed light on why that approach was taken) is to change the assumptions in the statement, how does it affect the approach in the proof, what no longer holds, is it even still true, what would you have to change in the proof for it to still work? Often it's only enlightening why a certain approach was taken, or why it's even a statement worth putting in the lecture notes, when you think beyond the original proposition. You also have to genuinely question everything with proofs, line by line, if you can't answer "Why?" to every step taken in a proof you're doing yourself a disservice (and making it harder in the long run). Once you start to properly understand approaches in proofs you essentially start to build up a 'problem solving' toolkit and become better at it.

It also helps just from the standpoint that a lot of problem solving is essentially just concatenating simpler ideas and proofs, so having a thorough knowledge of all the material makes your life easier. There are some proofs which require a spark of unintuitive ingenuity that you wouldn't otherwise get without essentially comprehensively knowing all the content.
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Hasufel
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reading and practice is a big part of it..familiarising yourself with the variety of techniques,

here`s a good sort of "into":

http://www.amazon.co.uk/Numbers-Proo.../dp/0340676531

also, there are plenty of papers on the internet about induction and direct proof.
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Smaug123
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(Original post by Brian Moser)
When asked to prove something, I'm usually clueless. Even given the necessary axioms to use, I never know what to do. However, once I've seen a proof I understand what they've done. This doesn't mean to say that reading lots of proofs improves my ability to do them myself: all I can do is memorise them and hope that only the exact same proofs are asked for in future.

Any advice on what would help my ability to do proofs?


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Most of the proofs you'll see have just one key idea. Any given topic has only around five key ideas to use.

For example, early Analysis has "continuous functions are ones which preserve limits of sequences", "examine what happens near a sup of a function", "use the Mean Value Theorem", "ab-cd = (a+c)b - (b-d)c", and "split the sum into positive terms and negative terms, and consider each separately". There's not that many ideas.

Can you recognise the key ideas in proofs you've seen? Can you reconstruct the proofs given the key ideas?
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