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How to attempt Apostol proofs

  1. Sep 13, 2008 #1
    1. The problem statement, all variables and given/known data
    If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x<z<y


    2. Relevant equations



    3. The attempt at a solution
    The problem arises from my inexperience in rigorously proving anything. If possible a general explanation of where to begin when trying to prove something rigorously would be more helpful than just the answer. I find that proofs generally seem obvious after i see them completed, but i sit for hours staring at them not knowing where to even remotely begin. Any help would be greatly appreciated. Thanks!
     
  2. jcsd
  3. Sep 13, 2008 #2

    morphism

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    There is no general algorithm for attacking problems. But usually it helps to think about what the problem is really asking. For instance, for the problem you've posted, think about this: can you find a number between 0 and 1? In general, what's the natural choice for something that sits in between two other things? (Maybe something that's 'in the middle'?)
     
  4. Sep 13, 2008 #3
    Ah! See, i feel really dumb now...haha...thanks. Hmmm...if i were to say something like, if x<y then (1/2)x is also less than y... would i then need to prove that? Or is that "obvious" enough to just simply state? I have a feeling that stating something as obvious is blasphemous in mathematics.
     
  5. Sep 13, 2008 #4

    morphism

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    But is (1/2)x always greater than x?
     
  6. Sep 13, 2008 #5
    Alright...thanks again! I think its just a matter of not over complicating things and thinking things through before trying to prove anything. Thanks!
     
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