602-bolzano–weierstrass

602-bolzano–weierstrass - a and b are...

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MATH 602, WINTER 2009 The Bolzano-Weierstrass theorem The Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. Here is a proof. The fact that a real number y has a decimal expansion n.d 1 d 2 d 3 . . . (where n is an integer and d j ∈ { 0 , 1 , . . . , 9 } , j = 1 , 2 , 3 , . . . ) amounts to the limit relation y j y as j → ∞ for the sequence (1) y j = n + 10 - 1 d 1 + 10 - 2 d 2 + . . . + 10 - j d j , j = 1 , 2 , 3 , . . . . We also set y 0 = n . For a given bounded sequence x j of real numbers, we will exhibit a limit y of a con- vergent subsequence t j of the sequence x j by explicitly constructing a decimal expansion (2) n.d 1 d 2 d 3 . . . of y , and making sure that, for all j = 0 , 1 , 2 , 3 , . . . and for y j as in (1), (3) | t j - y j | ≤ 10 - j . The fact that t j y , for the number y with the decimal expansion (1), will then be immediate, since y j y , while (3) gives t j - y j 0. Since x j is bounded, all the terms x j lie in some half-open interval [ a, b ). Making a smaller and b larger, we may assume that both
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Unformatted text preview: a and b are integers, and then partition [ a, b ) into the disjoint union of half-open intervals [ n, n +1), n = a, a +1 , . . . , b-1. For at least one such n , the subinterval [ n, n + 1) contains infinitely many terms x j (the precise meaning of which is, as usual, contains x j for infinitely many positive integers j ). Let us fix the smallest n ∈ { a, a + 1 , . . . , b-1 } with the property just named. This will be the integer part of our decimal expansion n.d 1 d 2 d 3 . . . . We also set t = x j (0) for the smallest positive integer j (0) with x j (0) ∈ [ n, n + 1). (Note that such j (0) exist, in fact, there are infinitely many of them.) Obviously, (3) now holds for j = 0. We now describe how to choose d j +1 and . ... further text in preparation...
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This note was uploaded on 11/08/2011 for the course MATH 601 taught by Professor Osu during the Fall '11 term at Cornell.

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