Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number.
Assume that there is a smallest negative rational number. Therefore, the number is of the form $r = - \frac{p}{q}$, where $p$ and $q$ are positive integers. But, there is a rational number $- \frac{p}{q+1}$, which is smaller than $r$. This is a contradiction. Therefore, there is no smallest negative rational number. QED
Assume that there is a largest positive rational number. Therefore, the number is of the form $r = \frac{p}{q+1}$, where $p$ and $q$ are positive integers. But, there is a rational number $\frac{p+1}{q}$, which is larger than $r$. This is a contradiction. Therefore, there is no largest positive rational number. QED
Do you think that these are correct proofs?