how to prove that a set is not bounded above?

Question

I got stuck in proving that $A=\{yn:n\in\mathbb N,y\in(1,\infty)\}$ is not bounded above? (without of course using lim)?

Answer 1

If $a \in A$ then $a = ny$ for some $n \ge 1$ and $y > 0$ so $a =ny>0$.

Suppose $M$ is an upper bound of $A$. Then for any $a \in A$ we have $M \ge a > 0$ so $M$ is a positive real number. So $M \in (0, \infty)$ and so $2M \in A$. But $M < 2M$ so $M$ can not be an upper bound.