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A funny proof, but what is wrong with it?

  1. Apr 10, 2012 #1
    Here's a question:-

    Prove that for all positive integers n,

    [(n+1)/2]^n >= n!

    And here's a funny proof for it:-

    Assume to the contrary that, for all positive integers n,

    [(n+1)/2]^n < n!

    However, for n=2,

    (3/2)^2 > 2!

    Therefore, our assumption must be false.

    And hence, for all positive integers n, [(n+1)/2]^n >= n!

    Now, I know that this proof can't be correct, because I've seen the real proof, and it's a marvel, making use of algebraic inequalities, and the proof above simply seems too simple compared to it. But I wonder, what's the mistake with the above proof?
     
  2. jcsd
  3. Apr 10, 2012 #2

    micromass

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    You've made a logic mistake.

    Consider the sentence

    "All men are mortal"

    This is true. But what's the negation of this sentence?? Is it

    "All men are immortal"?

    or is it

    "There is a man that is immortal"??

    It's the second one.

    So, if your assertion is

    "For all natural numbers n holds the inequality blabla"

    then the negation of that is

    "There is a natural number n such that the inequality does not hold"

    Your proof shows that the inequality holds for n=2. But this does not contradict the negation, as there could still be an n such that it doesn't hold.
     
  4. Apr 10, 2012 #3
    The assumption that for all possible integers n, n!>[(n+1)/2]^n is violated by your counter example, so that demonstrates that your assumption is false. You have not checked all possible values of n so it says nothing about any of those values. Your assumption being false says nothing about the case of only some n's satisfying the inequality.
     
  5. Apr 11, 2012 #4
    Oh...now I understand. Thanks, micromass and bmxicle! :D
     
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