## A.4 Theorems and Lemmas The dictionary defines a ***theorem*** as a proposition that sets forth something to be proved. It has the same meaning in mathematics. Each of the examples in the preceding section could be stated as a theorem, whereas the induction proof constitutes a proof of the theorem. For example, we could state [Example A.1](LiB0095.html#1293) as follows: Theorem A.1****For all integers *n* > 0, we have  Proof: The proof would go here. In this case, it would be the induction proof used in [Example A.1](LiB0095.html#1293). **** Usually the purpose of stating and proving a theorem is to obtain a general result that can be applied to many specific cases. For example, we can use [Theorem A.1](#ap-aex06) to quickly calculate the sum of the first *n* integers for any positive integer *n*. Sometimes students have difficulty understanding the difference between a theorem that is an "if" statement and one that is an "if and only if" statement. The following two theorems illustrate this difference. Theorem A.2****For any real number *x*, if *x* > 0, then *x*2 > 0. Proof: The theorem follows from the fact that the product of two positive numbers is positive. **** The reverse of the implication stated in [Theorem A.2](#ap-aex07) is not true. That is, it is not true that if *x*2 > 0, then *x* > 0. For example,  and −3 is not greater than 0. Indeed, the square of any negative number is greater than 0. Therefore, [Theorem A.2](#ap-aex07) is an example of an "if" statement. When the reverse implication is also true, the theorem is an "if and only if" statement, and it is necessary to prove both the implication and the reverse implication. The following theorem is an example of an "if and only if" statement. Theorem A.3****For any real number *x*, *x* > 0 if and only if 1/*x* > 0. Proof: Prove the implication Suppose that *x* > 0 Then  because the quotient of two positive numbers is greater than 0. Prove the reverse implication Suppose that 1/*x* > 0. Then  again because the quotient of two positive numbers is greater than 0 **** The dictionary defines a ***lemma*** as a subsidiary proposition employed to prove another proposition. Like a theorem, a lemma is a proposition that sets forth something to be proved. However, we usually do not care about the proposition in the lemma for its own sake. Rather, when the proof of a theorem relies on the truth of one or more auxiliary propositions, we often state and prove lemmas concerning those propositions. We then employ those lemmas to prove the theorem.