Yes, you need to prove that . Reasoning by contradiction, you assume that the negation is true. By definition of big-O, this means that there are , such that . Note that when you were proving , it was your task to find and to prove . Now, you assumed the negation, so those and that supposedly validate are given to you. You, as a state attorney, only need to take this evidence provided by the defense, and say, "Ladies and gentlemen of the jury, the argument that you've just heard is absurd. Indeed, plug and see for yourselves."The second one, I am trying a proof by contradiction (though I was always told to assume the negation of the hypothesis).