Cauchy Sequences
In dealing with real numbers, we assume that each finite and infinite decimal expansion defines a real number. When two numbers differ by [1/2] ·10-k > 0, their decimal expansions are said to agree to k decimal places. Convergence of a sequence to a limit L can now be expressed in terms of decimal numbers or significant digits: For any whole number k, there is a whole number N, such that all terms in the sequence after the first N agree with the limit L to k decimal places. Convergence here corresponds to the ability in principle, if not in practice, to patiently compute a decimal or binary expansion to an unlimited number of places. Error control in practice requires a rate of convergence estimate to say how large N must be to obtain k decimal places. We may distinguish between convergence arguments which says there is always N and convergence arguments which give N as an easily-computed function of k - convergence in principle versus the desired situation in which the rate of convergence can be described and computed. A Cauchy sequence f(n) has the following property: For each whole number k, there is a whole number N with the following property: all terms in the sequence after the first N-1 agree with each other to at least k decimal places. This property allows us to define and compute in principle an infinite decimal expansion. This expansion is assumed to define a unique real number: the limit L of the Cauchy sequence.
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Units in Calculations: Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area. |
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