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Homework Help: Cauchy sequence and convergeant diameters.

  1. May 5, 2013 #1
    Suppose (an) is sequence in the metric space X and define Tn={ak:k>n} and diamT=sup{d(a,b):a,b elements of T}.

    Prove that (an) is Cauchy if and only if diam Tn converges to zero.

    In what metric spacee does Tn converge? I assumed in (ℝ,de) but this is confusing since the diam of T is measured using the metric of X.
  2. jcsd
  3. May 5, 2013 #2
    ##T_n## is set in space X.

    ##\mathrm{diam} \ T_n## are real numbers, and nothing is said what metric is to be used on them. Which probably means the "natural" Euclidean metric.
  4. May 5, 2013 #3
    That makes sense, thanks.
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