[PDF] cauchy sequence example

For example, the sequence 1, 1/2, 1/3, 1/4, is Cauchy in the metric space of real numbers, because for any epsilon, we can choose N to be larger than 1/epsilon, and then the distance between any two terms after the N-th term will be less than 1/N, which is less than epsilon.
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  • What is an example of a Cauchy sequence which is convergent?

    The sequence {Xn} defined by X1=?2 and X(n+1)=?(2Xn) for all n>=1(converges to 2) is another example of a Cauchy Sequence.

  • Which is the Cauchy sequence?

    A Cauchy sequence is a sequence in which the difference between any two terms becomes arbitrarily small as the index of the terms increases.
    In other words, we define the Cauchy sequence as a sequence of real or complex numbers that converge to a limit in a metric space.5 mai 2023

  • How do you know if a sequence is Cauchy?

    1.
    A sequence {an}is called a Cauchy sequence if for any given ? > 0, there exists N ? N such that n, m ? N =? an ? am < ?. an ? L < ? 2 ? n ? N.
    Thus if n, m ? N, we have an ? am?an ? L + am ? L < ? 2 + ? 2 = ?.

  • How do you know if a sequence is Cauchy?

    We have a Cauchy Sequence which is not convergent.
    For Example: The sequence 1, 1/2, 1/3, 1/4, 1/5,

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