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Complete Metric Spaces: Prove that the intersection is nonempty.

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Complete Metric Spaces.

Problem 4: By the diameter of a subset A of a metric space R is meant the number

d(A) = sup p(x,y) where x,y are real numbers of A

Suppose R is complete, and let {A_n} be a sequence of closed subsets of R nested in the sense that

a_1 is a superset of A_2 is a superset of .... A_4 is a superset of ...

Suppose further that

lim as n ---> infinity of d(A_n) = 0.

Prove that the intersection of infinity, n = 1 of A_n is nonempty.

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Explanations:

The proof is similar to the proof of Theorem 2 on page 60, with a suitable modification.

Let x_n∈A_n be a ...

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