If you think you can disprove x > y, (where x and y are parameters like the mean of some quantity for two different populations) then it follows that you can prove x <= y.
On the other hand you cannot ever prove x = y since there will always be values of x and y that are so close that your test has no power. but you can still put bounds on how close x and y must be in this case
In this case I think both sides are right, so the argument should be used in context. You can only prove things about the real world (and not an axiomatic system for example) assuming there are no unknowns. In some (perhaps most) cases those unknowns are not there, but in some important cases unexpected behavior may occur in spite of "overwhelming evidence", because the assumptions made by the conclusions failed. It would be a much simpler discussion to state "When you want to prove something, make your assumptions clear, and test them well. Everything that hopes for rigor is proved within a well defined set of assumptions, so it's not like nothing can be proved either.".
If they are so close that your test has no power, then they might as well be the same for the purposes of your test.