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Unformatted text preview: CONSTRUCTION OF R 1. MOTIVATION We are used to thinking of real numbers as successive approximations. For example, we write = 3 . 14159 ... to mean that is a real number which, accurate to 5 decimal places, equals the above string. To be precise, this means that   3 . 14159  < 1 2 10 5 . But this does not tell us what is . If we want a more accurate approximation, we can calculate one; to 10 decimal places, we have = 3 . 1415926536 ... Continuing, we will develop a sequence of rational approximations to . One such sequence is 3 , 3 . 1 , 3 . 14 , 3 . 142 , 3 . 1416 , 3 . 14159 ,... But this is not the only sequence of rational numbers that approximate closer and closer (far from it!). For example, here is another (important) one: 3 , 22 7 , 333 106 , 355 113 , 104348 33215 , 1043835 332263 ... (This sequence, which represents the continued fractions approximations to [you should read about continued fractions for fun!], gives accuracies of 1 , 3 , 5 , 6 , 9 , 9 decimal digits, as you can readily check.) Mores the point, here is another (somewhat arbitrary) rational approximating sequence to : , , 157 , 45 , 10 , 3 . 14159 , , 3 . 14 , 3 . 1415 , 3 . 151592 , 3 . 14159265 ,... While here we start with a string of numbers that really have nothing to do with , the sequence eventually settles down to approximate closer and closer (this time by an additional 2 decimal places each step). The point is, just what the sequence does for any initial segment of time is irrelevant; all that matters is what happens to the tail of the sequence. We are going to use the above insights to actually give a construction of the real numbers R from the rational numbers Q . The idea is, a real number is a sequence of rational ap proximations. But we have to be careful since, as we saw above, very different sequences of rational numbers can equally well approximate the same real number. To take care of this ambiguity, we will have to define real numbers as sets of rational approximating sequences, all with the same tail behaviour . To make this precise, the following section contains precise definitions and some theorems which we will need to set out the formal mathematical construction of R (as envisioned by Bolzano and Cauchy in the early 19th century). 2. CAUCHY SEQUENCES We have already been working with sequences, but to be sure were on the same page: Definition 1. A sequence of rational numbers (aka a rational sequence ) is a function from the natural numbers N into the rational numbers Q . That is, it is an assignment of a rational number to each natural number. We usually denote such a function by n 7 a n , so the terms in the sequence are written { a 1 ,a 2 ,a 3 ,... } To refer to the whole sequence, we will write ( a n ) n =1 , or for the sake of brevity simply ( a n ) ....
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 Fall '10
 Prof.KatrinWehrheim
 Real Numbers, Approximation

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