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Can't prove things to save my life HELP Watch

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1. I absolutely hate proofs, I never know how to start them, I don't know whats useful to say or if I'm on the right track. I hoped my university would teach me how to do these kind of questions but I'm getting assignments where I'm expected to prove something, I could just google search a start but then I would be at the same problem for the next assignment, and god forbid I need to prove something in an exam.

I need a general method, everything online has been useless and vague so far, for example how do I proof sqrt(ab) < 1/2(a+b) for arithmetic and geometric means. I tried writing some statements such as a>0 , b>0 a+b >0, ab>0 but I just have nothing new to write and get stuck.

Help appreciated.
2. practice helps, and a few ideas of where to start. For example with that one, I'd start by ASSUMING it is correct, trying some things out to manipulate the equation, and see if something that is obviously true can be derived from it. Then you can do the exact opposite process, starting from the thing you know to be true, and derive the equation again.

In the case of sqrt(ab) < (a+b)/2, a tempting first step is to square both sides to get rid of the sqrt. That frees you up to do many more operations. Then just try manipulating things and see where it takes you. (maybe 4 or 5 steps should do it )
3. (Original post by mik1a)
practice helps, and a few ideas of where to start. For example with that one, I'd start by ASSUMING it is correct, trying some things out to manipulate the equation, and see if something that is obviously true can be derived from it. Then you can do the exact opposite process, starting from the thing you know to be true, and derive the equation again.

In the case of sqrt(ab) < (a+b)/2, a tempting first step is to square both sides to get rid of the sqrt. That frees you up to do many more operations. Then just try manipulating things and see where it takes you. (maybe 4 or 5 steps should do it )
Its hard to practise something you cant really do, do you have a general method or steps you follow?

Ill try the manipulation thing but is it a "proper" and rigourous way to show something.

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Updated: October 9, 2016
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