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  3. xercise 162 used in example 163 let t be the...

Question: xercise 162 used in example 163 let t be the...

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xercise 1.6.2. [Used in Example 1.6.3.] Let T be the set defined in Equation L Used in Example 1.o.. (1) Prove that T is a Dedekind cut.36 1 Construction of the Real Numbers out to be a simple solution to this problem, as we will now see. We note first, however that we have not yet formally defined what V2 means, nor proved that there is such a real number, though we will do so in Theorem 2.6.9 and Definition 2.6.10. We have also not yet proved that v2 is not rational, a fact with which the reader is, at least informally, familiar; we will see a proof of this fact in Theorem 2.6.11. More precisely, it will be seen in that example that there is no rational number x such that xi = 2, and this last statement makes use only of rational numbers, so it is suited to our purpose at present. Nothing in our subsequent treatment ofV2 in Section 2.6 makes use of the current example, so it will not be circular reasoning for us to make use of these subsequently proved facts here. Let T={x E Q | x > 0 and x2 >2) It is seen by Exercise 1.6.2 (1) that T is a Dedekind cut, and by Part (2) of that exercise it is seen that if T has the form (xEQIx>r] for some r E Q, then r2-2. By Theorem 2.6.11 we know that there is no rational number x such that x2 2, and it follows that T is a Dedekind cut that is not of the form given in Lemma 1.6.2. Example 1.6.3 explains the need for the following definition.

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