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Math Help - Well-Ordering

  1. #1
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    Well-Ordering

    Prove: If y>0 then there exists a natural number, n, such that n-1 is less than or equal to y < n.

    I'm supposed to prove using the well-ordering property on the set {m as an element of the natural numbers : m > y}
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  2. #2
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    Quote Originally Posted by GoldendoodleMom View Post
    Prove: If y>0 then there exists a natural number, n, such that n-1 is less than or equal to y < n.
    I'm supposed to prove using the well-ordering property on the set {m as an element of the natural numbers : m > y}
    To use that, you must know that \mathbb{N} is not bounded above.
    Because that is true, T = \left\{ {m \in \mathbb{N}:y < m} \right\} \ne \emptyset .
    Every nonempty subset of \mathbb{N} has a first term.
    Let n = \min \left( T \right) \Rightarrow n \in T \Rightarrow y < n.
    Moreover, because n - 1 < n\quad  \Rightarrow \quad n - 1 \notin T\quad  \Rightarrow \quad n - 1 \leqslant y < n.
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