# Math Help - Proving sets unbounded

1. ## Proving sets unbounded

Prove that for every natural number n, the set { m/n :m is a natural numer} is unbounded above in R.

2. Originally Posted by Jdg6057
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.