Chap 1 The Number System

The real number m satisfying conditions i1 and i2 is

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Unformatted text preview: xists x ∈ S such that x &gt; 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 non-empty 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 &gt; 0, there exists x ∈ S such that x &lt; m + . Definition. The real number M satisfying conditions (S1) and (S2) is called the supremum of the non-empty set S , and denoted by M = sup S . The real number m satisfying conditions (I1) and (I2) is called the inﬁmum of the non-empty 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 &lt; 2}. Clearly the set S is non-empty, 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.

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