C.4 Practice Proof 1

Now read this short theorem and proof and self-explain each line, either in your head or by making notes on a piece of paper, using the advice from the preceding pages.

Theorem.There is no smallest positive real number.

Proof. Assume, to the contrary, that there exists a smallest positive real number.

Therefore, by assumption, there exists a real number $r$ such that for every positive number $s$, $0<r<s$.

Consider $m=\frac{r}{2}$.

Clearly, $0<m<r$.

This is a contradiction because $m$ is a positive real number that is smaller than $r$.

Thus there is no smallest positive real number. $\square$