Some articles on cauchy sequences, sequences, cauchy, cauchy sequence, sequence:
Absolute Value (algebra) - Completions
... Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 there is a positive integer N such that for ... It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a ring ... One can also define null sequences as sequences of elements of D such that
... Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 there is a positive integer N such that for ... It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a ring ... One can also define null sequences as sequences of elements of D such that
Cauchy Sequence - Generalizations - In Constructive Mathematics
... In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful ... If is a Cauchy sequence in the set, then a modulus of Cauchy convergence for the sequence is a function from the set of natural numbers to itself, such that ... Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence ...
... In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful ... If is a Cauchy sequence in the set, then a modulus of Cauchy convergence for the sequence is a function from the set of natural numbers to itself, such that ... Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence ...
Complete Metric Space - Completion
... constructed as a set of equivalence classes of Cauchy sequences in M ... For any two Cauchy sequences (xn)n and (yn)n in M, we may define their distance as (This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric ... distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M ...
... constructed as a set of equivalence classes of Cauchy sequences in M ... For any two Cauchy sequences (xn)n and (yn)n in M, we may define their distance as (This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric ... distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M ...
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