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?

If $0 < \epsilon \in \Bbb R$, and $\epsilon < r$ for every positive $r \in \Bbb R$, then
$\epsilon < \epsilon; \tag 1$
that will obviously not work!
Even if we relax the condition from $\epsilon < r$ to $\epsilon \le r$ for all positive $r \in \Bbb R$, it still won't work, since
$0 < \dfrac{\epsilon}{2} < \epsilon. \tag 2$
