cauchy sequences การใช้

• In the real numbers every Cauchy sequence converges to some limit.
• In these cases, the concept of a Cauchy sequence is useful.
• In metric spaces, one can define Cauchy sequences.
• In a similar way one can define Cauchy sequences of rational or complex numbers.
• Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence.
• In a general metric space, however, a Cauchy sequence need not converge.
• By definition, in a Hilbert space any Cauchy sequence converges to a limit.
• The usual decimal notation can be translated to Cauchy sequences in a natural way.
• Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
• Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.
• Weakly Cauchy sequences in are weakly convergent, since-spaces are weakly sequentially complete.
• Completeness can be proved in a similar way to the construction from the Cauchy sequences.
• Every Cauchy sequence is bounded, although Cauchy nets or Cauchy filters may not be bounded.
• This obviously defines two Cauchy sequences of rationals, and so we have real numbers and.
• Instead of working with Cauchy sequences, one works with Cauchy filters ( or Cauchy nets ).
• The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.
• The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
• The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use.
• Since the metric space is complete this Cauchy sequence converges to some point " x ".
• A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric.