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Math Help - Proving sets unbounded

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    Proving sets unbounded

    Prove that for every natural number n, the set { m/n :m is a natural numer} is unbounded above in R.
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    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Jdg6057 View Post
    Prove that for every natural number n, the set { m/n :m is a natural numer} is unbounded above in R.
    Prove: \forall~n\in\mathbb{N}, A=\left\{\frac{m}{n}: m\in\mathbb{N}\right\} is unbounded above in \mathbb{R}.

    Proof: Fix n\in\mathbb{N}. Assume A is bounded; i.e. \exists~x\in\mathbb{R} such that \frac{m}{n}\leq x, ~\forall~ m\in\mathbb{N}. This implies that m\leq nx, ~\forall~ m\in\mathbb{N}. Because nx is a constant, though, the previous statement implies that \mathbb{N} is bounded above by nx, which is of course a ridiculous statement. Thus we have a contradiction and A is unbounded above.
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