Show that there is no positive real number which is less than every positive real number. I think what would happen if there is less than every positive real numbers
How can I prove that?

Let $r$ be a positive real. Consider the set $S=\{x\in\mathbb{R}~:~x<r\}$. Now this set $S$ is non-empty, as $0\in S$. So it has a lowest upper bound, say $r^*$. $r^*\leq r$ as $r\not<r$. If $r^*<r$, then $r^*<\frac{r+r^*}{2}<r$. Hence $\frac{r+r^*}{2}\in S$, which contradicts our assumption that $r^*$ is the lowest upper bound. So $r^*=r$, but $r\not\in S$.
