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Math Help - Prove by induction

  1. #1
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    Prove by induction

    1. Prove by induction:
    (a) 1 \le n\ \forall n \in N;
    (b)  \forall n \in N, either n = 1 or n-1 \in N; and
    (c) \forall n \in N, there is no element m of N in the range n < m < n+1.

    2. Prove by induction on n that \forall m, n \in N\ \exists q \in N such that  qm > n.
    Also prove that for every n \in N, there is an m \in N such that  2^m > n .
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  2. #2
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    Quote Originally Posted by mathwizard View Post
    1. Prove by induction:
    (a) 1 \le n\ \forall n \in N;
    (b)  \forall n \in N, either n = 1 or n-1 \in N; and
    (c) \forall n \in N, there is no element m of N in the range n < m < n+1.

    2. Prove by induction on n that \forall m, n \in N\ \exists q \in N such that  qm > n.
    Also prove that for every n \in N, there is an m \in N such that  2^m > n .
    You can prove some of these without induction, if that is okay with you.
    Like Problem 1(c), you can do it by definition of what it means being a natural number.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    You can prove some of these without induction, if that is okay with you.
    Like Problem 1(c), you can do it by definition of what it means being a natural number.
    No. The question specifies to prove the statements by induction. I think the proof relies on the four axioms that a field F must satisfy. This is from my advanced calculus textbook (first chapter & first section, btw).
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