Proof a+b>2ab a2+2ab+62>4ab a2-2ab+b2>0 The last statement is true because the square of a real number(such as a-b)is never negative. This proves the claim In this argument, we started with what we wanted to prove and then reasoned until we reached a statement that is surely true. The little question marks, I guess, are supposed to indicate that were not quite certain that the inequalities are valid until we get down te the last step. At that point, we know everything checks out, apparently 3.1 Why reasoning backward Is bad In reasoning backward, we began with the proposition in question-call it P-and rea- soned to a true conclusion. Thus, what we actually proved is P→true But this implication is trivially true, regardless of whether P is true or false! Therefore, by rea soning backward we can"prove"not only true statements but also every false statement! Heres an exampl Claim.0=1 Proof 0=1 2 So resist the urge to reason backward. If this keeps happening to you anyway, pound your writing hand with a heavy textbook to make it stop Shout"WRONG! WRONG! with each blow8 Proofs Proof. a + b 2 a + b a 2 + 2ab + b2 ? ≥ √ ab ? √ ≥ 2 ab ? ≥ 4ab ? a ≥ 0 2 − 2ab + b2 (a − b) 2 ≥ 0 The last statement is true because the square of a real number (such as a − b) is never negative. This proves the claim. × In this argument, we started with what we wanted to prove and then reasoned until we reached a statement that is surely true. The little question marks, I guess, are supposed to indicate that we’re not quite certain that the inequalities are valid until we get down to the last step. At that point, we know everything checks out, apparently. 3.1 Why Reasoning Backward Is Bad In reasoning backward, we began with the proposition in question— call it P— and rea￾soned to a true conclusion. Thus, what we actually proved is: P ⇒ true But this implication is trivially true, regardless of whether P is true or false! Therefore, by rea￾soning backward we can “prove” not only true statements, but also every false statement! Here’s an example: Claim. 0 = 1 Proof. 0 0 · 0 ? = 1 ? = 1 · 0 0 = 0 × So resist the urge to reason backward. If this keeps happening to you anyway, pound your writing hand with a heavy textbook to make it stop. Shout “WRONG! WRONG!” with each blow