I don't like the conclusion that students should avoid proving by contradiction unless they really need it. I almost always start with contradiction as my init

I don't like the conclusion that students should avoid proving by contradiction unless they really need it. I almost always start with contradiction as my initial method of choice when taking exams because you don't always have time to completely rewrite your proofs. Contradiction helps with this since it naturally encompasses proving by contrapositive also. Because if you're trying to prove that p->q, you can assume p and ~q and getting to ~p is a contradiction since p^~p (and this is the same structure as a proof by contrapositive if you got to this line without assuming p). And of course if you find any other absurdities along the way sooner, you're done quicker. This is nice because a proof by contrapositive is more or less equivalent to a direct proof. So simply by always starting with proof by contradiction, you cover the cases where the theorem: - is simple enough for you to find out exactly why it's true (direct proof hidden in the contrapositive). - is complex and wide reaching enough for you to hit an absurdity quickly