Cauchy sequence and convergeant diameters.

[gottfried]

Suppose (a_n) is sequence in the metric space X and define T_n={a_k:k>n} and diamT=sup{d(a,b):a,b elements of T}.

Prove that (a_n) is Cauchy if and only if diam T_n converges to zero.

In what metric spacee does T_n converge? I assumed in (ℝ,d_e) but this is confusing since the diam of T is measured using the metric of X.

 

[voko]

$T_n$ 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.

 

[gottfried]

That makes sense, thanks.