TITLE: Show that there is no positive real number which is less than every positive real number.
QUESTION [5 upvotes]: I think what would happen if there is less than every positive real numbers
How can I prove that?

REPLY [10 votes]: Suppose that there does exist some real $y > 0$ that is less than every other positive real number. Then $y < \frac{1}{n}$ for every natural number $n$. This would then imply that $n < \frac{1}{y}$ for every natural number (the natural numbers would then have an upper bound!) and so we have a contradiction.