In these cases, the concept of a Cauchy sequence is useful. The use of Cauchy sequences is one of the two famous ways of defining the real numbers, that is, completing the rationals. Examples In the real numbers every Cauchy sequence converges to some limit. The reals are the metric completion of the rationals. Metrically complete means that every Cauchy sequence made from the set converges to an element which is itself in the set. We have defined an extension to the rationals that is metrically complete-that extension of the rationals is the real numbers. Once we have done that, the payoff is enormous.
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