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Induction proof of an inequality

  1. May 8, 2010 #1
    1. The problem statement, all variables and given/known data

    for all integers n>=1, n! <= n^n

    2. Relevant equations

    3. The attempt at a solution

    Base case: (1)! <= (1)^(1) 1=1 check
    Inductive hypothesis: suppose k!<=k^k
    P(k+1): (k+1)! <= (k+1)^(k+1)

    From here on out I get very confused. Any help would be appreciated!
  2. jcsd
  3. May 8, 2010 #2


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    Write [itex](k+1)! \le (k+1)^{k+1}[/itex] in terms of k and k^k.
  4. May 8, 2010 #3
    so it would be k!(k+1) <= (k+1)^k + (k+1) ?
  5. May 8, 2010 #4


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    The right hand side is incorrect, but you're on the right track.
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