There is no smallest rational number greater than 2

1,483

Suppose there was a smallest rational number greater than $2$. Call it $k =p/q$.

Then consider $ k' = \frac{2+k}{2}$. This is a number bigger than $2$ and less than $k$. Also $k'$ is rational. Therefore there is no smallest rational number greater than $2$.

Share:
1,483
mm19
Author by

mm19

hi there!

Updated on April 25, 2020

Comments

  • mm19
    mm19 over 3 years

    I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why.

    Prove the statement “There is no smallest rational number greater than 2” by contradiction.

  • mm19
    mm19 almost 8 years
    Thank you I understand what your saying. I think I'll be able to write something up now.