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| Proof:
First, by mathematical induction, n < j implies qn ³ qj. Second, if there exist a whole number K > 0 such that k ³ K implies qk = qK then the limit is given by A = qK and the sequence takes on at most K distinct values q1,¼,qK. Otherwise for every K > 0, there is a k > K such that qk ¹ qK. In this case, the range of values assumed by the sequence is an infinite set. The case where the range of values is an infinite set is considered next. An small assertion: observe A > qm for some m implies that A > qm ³ qn for all n ³ m. This in turn implies A is not a limit point. The interval of length qm-A centered at A contains at most m elements of the sequence. That A > qm for some m implies that A is not a limit point of the sequence qj ³ qj+1. According to the Weierstrass theorem, the infinite set formed by the sequence qj has a limit point A. The contrapositive of the above assertion now implies that the limit point A £ qp for all whole numbers p. But now for each whole number k, the interval A+[1/2]10-k to A (with A included) contains infinitely many elements of the sequence. Therefore there is some sequence element qn with the property that A+[1/2]10-k ³ qn ³ A. Therefore p ³ n implies A+[1/2]10-k ³ qn ³ qp ³ A since qp ³ A and qn ³ qp whenever p > n. Finally, since k is arbitrary, we conclude include the sequence qn converges to a limit A. Moreover, we can say there are no points in the sequence qn to the left of A. |
Theorem D.3 [On Increasing Sequences]
If the infinite sequence q1,q2,q3¼
has the property that for each whole numbers k, qk
£ qk+1 £
Q for some real number Q, an upper bound, then
the sequence converges to a limit B £
Q. The limit B is the least upper bound of the
set of points in the sequence.
the Real Analysis appendices of
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Presenting Appendices from Volume 3, Why Slopes and More Math, If the epsilon-delta viewpoint of limits, continuity and convergence is not yet comfortable, see Chapters 14 to 19 in Volume 3 are related.
Proofs of one-sided theorems could be of interest in the study of 2D topology.
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If you like these appendices to Volume 3, you may also like (a) the foreword of Volume 3 and chapter 14 with its decimal view of limits, (b) Volume 2, Three Skills for Algebra (for its 4 skills, not 3, for algebra), (c) this treatment of Exponents & Radicals Exactly, (d) this geometric treatment of complex numbers, (e) the Euclidean Geometry with a geometric proof of the distributive law for complex numbers, (f) Pattern Based Reason - its logic elements and online postscripts for |
Vol 1A Logic Postscripts
online only include
Proof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
Reality versus or with the aid of Imagination
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