A student claims that $11000\displaystyle \frac{1}{1000}$ is the smallest positive rational number.

Their teacher says that there is no smallest positive rational number, and gives the following correct proof.

Step 1: Assume there is a smallest positive rational number, and let this number be $r$ .

Step 2: Let $\dfrac{r}{2}$ .

Step 3: Then $s\,$ is rational and $0 < s < r$ .

Step 4: Therefore there is no smallest positive rational number.

Explain why Step 3 gives a contradiction.

Use a similar method to prove by contradiction that there is no largest negative rational number.

A student claims that $11000\displaystyle \frac{1}{1000}$ is the smallest positive rational number.

Their teacher says that there is no smallest positive rational number, and gives the following correct proof.

Step 1: Assume there is a smallest positive rational number, and let this number be $r$ .

Step 2: Let $\dfrac{r}{2}$ .

Step 3: Then $s\,$ is rational and $0 < s < r$ .

Step 4: Therefore there is no smallest positive rational number.

Explain why Step 3 gives a contradiction.

Use a similar method to prove by contradiction that there is no largest negative rational number.

1 Pure Mathematics

1 Pure Mathematics Questions