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Thread: 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: $\displaystyle \forall~n\in\mathbb{N}$, $\displaystyle A=\left\{\frac{m}{n}: m\in\mathbb{N}\right\}$ is unbounded above in $\displaystyle \mathbb{R}$.

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