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Math Help - Proof using Well Ordering

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    Proof using Well Ordering



    This is a real baby proof, but I'm not sure I'm doing the well ordering proof right, as its my first attempt.

    Is this the basic structure of a well ordering proof?

    Thanks in advance.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post

    This is a real baby proof, but I'm not sure I'm doing the well ordering proof right, as its my first attempt.

    Is this the basic structure of a well ordering proof?

    Thanks in advance.
    I feel as though you are way overdoing it. Let S=\left\{n\in\mathbb{N}:a^{n+1}\ne a\cdot a^n\right\}. Assume that S\ne\varnothing, then by the well-ordering principle S has a least element k. Thus, a^{k+1}\ne a^k\cdot a. Dividing both sides we see that a^{k}\ne a^{k-1}\cdot a\implies k-1\in S. This contradicts the minimality of k and thus it follows that S=\varnothing.


    It needs some preening, but you get the idea.
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