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Existence of a natural number X

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Given [itex]M \in N[/itex], show that there exists an [itex] X \in N [/itex] such that for all [itex] n \geq X [/itex], [itex] n^2+n+1 \succ M [/itex]

    2. Relevant equations

    3. The attempt at a solution
    Since both M and X are natural numbers and I am just trying to prove the existence of a certain natural number X, I thought that i could just set X = M.
    Then, n^2+n+1 [itex]\geq[/itex] X^2+X+1 since n [itex]\geq[/itex] X.
    And X^2+X+1=M^2+M+1[itex]\succ[/itex]M.
    So, n^2+n+1[itex]\succ[/itex]M.
    Is this a sufficient proof for the existence of X?
    It just doesn't feel like a full proof, should X be more limited?
  2. jcsd
  3. Nov 6, 2011 #2


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    If that is the correct statement of the problem, I don't see anything wrong with your argument. But I have an uneasy feeling like you do. Since it seems so trivial I wonder if the original problem is mis-copied or misunderstood.
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