I am looking for some examples of when proof by contradiction is used in a problem with more than one case.
In all the elementary examples, there are only two options (eg rational/irrational, infinite/finite), so you assume the opposite, show it cannot be true and then conclude the result.
However, I remember from my undergraduate study situations where there were three or more options and proof by contradiction was used to eliminate one or more of these.
I do not remember any of these examples anymore, but I was wondering if I could have some examples of statements (ideally elementary, but more advanced works) that follow such a style in their proof. This would be good to have to show students I teach typical contexts in which this may arise.