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.