Question on "Proof by contradiction" I am a little confused whether the following set of arguments can constitute a proof by contradiction:
Required to prove: Statement A is true.
Assumption: Suppose statement A is false.
We then show that if A is false then a (different) statement B is true.
But, statement B is true implies A is true.
Hence, A must be true.
These set of arguments feel a bit bizzare to me and perhaps might be incorrect. I am unable find a mistake.

there isn't a smallest rational number greater than zero.
Assumption: there is a smallest rational number greater than zero.
So, call him $r$.
Now you can take $r/2$.
So B: given a rational number $r$ greater than zero, there exists another rational number $r/2$ that is greater than zero and smaller than $r$.
So A.
