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Unformatted text preview: xists x ∈ S such that x > M − . Remark. It is not diﬃcult to prove that the number M above is unique. It is also easy to deduce that if S is a nonempty set of real numbers and S is bounded below, then there is a unique real number m ∈ R satisfying the following two conditions: (I1) For every x ∈ S , the inequality x ≥ m holds. (I2) For every > 0, there exists x ∈ S such that x < m + . Definition. The real number M satisfying conditions (S1) and (S2) is called the supremum of the nonempty set S , and denoted by M = sup S . The real number m satisfying conditions (I1) and (I2) is called the inﬁmum of the nonempty set S , and denoted by m = inf S . √ Let us now try to understand how numbers like 2 ﬁts into this setting. Recall that there is √ no rational number which satisﬁes the equation x2 = 2. This means that the number that we know as 2 is not a rational number. We now want to show that it is a real number. Let S = {x ∈ R : x2 < 2}. Clearly the set S is nonempty, since 0 ∈ S . On the other hand, the se...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math, Calculus

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