# how to prove that a set is not bounded above?

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)?

• Suppose $M$ was its upper bound. What can you do with that? – John Douma Apr 10 '20 at 20:34
• Well.... do you know how to show $(1,\infty)$ is not bounded above? If $y\in (1,\infty)$ and $n \in \mathbb N$ then $ny \ge y$. – fleablood Apr 10 '20 at 21:11
• Assume bounded . Archimedean principle? Then? – Peter Szilas Apr 10 '20 at 21:12
• If $a \in A$ then $a > 0$ and if $M \ge a$ then $M > 0$ so $2M \in A$. But $M < 2M$. So $M$ can not be an upper bound. – fleablood Apr 10 '20 at 21:26

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.