To achieve this the steps in a proof must follow logically from previous steps

To achieve this the steps in a proof must follow

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the proposition in question. To achieve this, the steps in a proof must follow logically from previous steps or be justified by some other agreed-upon set of facts. In addition to being valid, these steps must also fit coherently together to form a cogent argument. Mathematics has a specialized vocabulary, to be sure,
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1.2. Some Preliminaries 9 but that does not exempt a good proof from being written in grammatically correct English. The one proof we have seen at this point (to Theorem 1.1.1) uses an indirect strategy called proof by contradiction . This powerful technique will be employed a number of times in our upcoming work. Nevertheless, most proofs are direct. (It also bears mentioning that using an indirect proof when a direct proof is available is generally considered bad manners.) A direct proof begins from some valid statement, most often taken from the theorem’s hypothesis, and then proceeds through rigorously logical deductions to a demonstration of the theorem’s conclusion. As we saw in Theorem 1.1.1, an indirect proof always begins by negating what it is we would like to prove. This is not always as easy to do as it may sound. The argument then proceeds until (hopefully) a logical contradiction with some other accepted fact is uncovered. Many times, this accepted fact is part of the hypothesis of the theorem. When the contradiction is with the theorem’s hypothesis, we technically have what is called a contrapositive proof. The next proposition illustrates a number of the issues just discussed and introduces a few more. Theorem 1.2.6. Two real numbers a and b are equal if and only if for every real number ϵ > 0 it follows that | a b | < ϵ . Proof. There are two key phrases in the statement of this proposition that warrant special attention. One is “for every,” which will be addressed in a moment. The other is “if and only if.” To say “if and only if” in mathematics is an economical way of stating that the proposition is true in two directions. In the forward direction, we must prove the statement: ( ) If a = b , then for every real number ϵ > 0 it follows that | a b | < ϵ . We must also prove the converse statement: ( ) If for every real number ϵ > 0 it follows that | a b | < ϵ , then we must have a = b . For the proof of the first statement, there is really not much to say. If a = b , then | a b | = 0, and so certainly | a b | < ϵ no matter what ϵ > 0 is chosen. For the second statement, we give a proof by contradiction. The conclusion of the proposition in this direction states that a = b , so we assume that a ̸ = b . Heading o ff in search of a contradiction brings us to a consideration of the phrase “for every ϵ > 0.” Some equivalent ways to state the hypothesis would be to say that “for all possible choices of ϵ > 0” or “no matter how ϵ > 0 is selected, it is always the case that | a b | < ϵ .” But assuming a ̸ = b (as we are doing at the moment), the choice of ϵ 0 = | a b | > 0 poses a serious problem. We are assuming that | a b | < ϵ is true for every ϵ > 0, so this must certainly be true of the particular ϵ 0 just defined. However,
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