By DAVID ALEXANDER BRANNAN
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Extra resources for A First Course in Mathematical Analysis
1, which is the correct answer. Similarly, we canp define the product of any two positive real numbers. 4142 .. 1415 .. 4427093 .. We do not expect you to use this method to add decimals! 1: Numbers 32 pﬃﬃﬃ As before, we define 2 Â p to be the least upper bound of the set of numbers in the right-hand column. Similar ideas can be used to define the operations of subtraction and division. Thus we can define arithmetic with real numbers in terms of the familiar arithmetic with rationals, using the Least Upper Bound Property of R.
Notice that, if M is an upper bound of E and M 2 E, then part 2 is automatically satisfied, and so M ¼ sup E ¼ max E. Example 4 Determine the least upper bound of E ¼ f1 À n12 : n ¼ 1; 2; . g. Solution We guess that the least upper bound of E is M ¼ 1. Certainly, 1 is an upper bound of E, since 1 1 À 2 1; for n ¼ 1; 2; . : n To check part 2 of the strategy, we need to show that, if M0 < 1, then there is some natural number n such that 1 1 À 2 > M0: (1) n However 1 1 1 À 2 > M0 , 1 À M0 > 2 n n 1 , < n2 ðsince 1 À M 0 > 0Þ 1rÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ M0 1 1 , >0
3 we have taken for granted the existence of rational powers and their properties, without giving formal definitions. pﬃﬃﬃ How can we supply these definitions? For example, how can we define 2 as a decimal? Consider the set È É E ¼ x 2 Q : x > 0; x2 < 2 : 29 The Least Upper Bound Property of R is an example of an existence theorem, one which asserts that a real number exists having a certain property. Analysis contains many such results which depend on the Least Upper Bound Property of R. While these results are often very general, and their proofs elegant, they do not always provide the most efficient methods of calculating good approximate values for the numbers in question.
A First Course in Mathematical Analysis by DAVID ALEXANDER BRANNAN