Explore BrainMass
Share

Cauchy Sequence and Completeness of a Metric Space

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Let X be a complete metric space. If F_n is a sequence of nonempty closed subsets of X such that F_n+1 is contained in F_n and the limit as n-->infinity of the diameter(F_n) = 0, show that the interesection of all F_n is nonempty.

© BrainMass Inc. brainmass.com March 21, 2019, 9:24 pm ad1c9bdddf
https://brainmass.com/math/algebra/cauchy-sequence-completeness-metric-space-374355

Solution Preview

First, let's write down the relevant definitiions.
A metric space X is complete if every Cauchy sequence {x_n} of elements of X has a limit x also in X.
A sequence x_n is Cauchy, if for every epsilon>0 there's a number N such that for all n,m>N we have
d(x_n , x_m) < epsilon
here d(x,y) is the metric on X.
Now, the diameter diam(F) of a set F in a metric space is sup {d(x,y), x and y are in F}.

What we want to do is to construct a Cauchy ...

Solution Summary

Cauchy sequence is clearly evaluated in this case.

$2.19