Re: Intersection of decreasing balls

Here is a counterexample:

Let N be the set of positive integers. Define a metric on N:
d(m,n)=1+ 1/(mn) if m =/= n, =0 if m=n.

Then the closed balls B(1,1+1/2), B(2, 1+1/3),...,
B(n, 1+1/(n+1)),..... are decreasing and have empty intersection.

In Banach space, what you said is correct.

TCL
.

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