I'm reading soviet textbook "Elements of theory of functions and functional analysis" by Kolmogorov and Fomin. There is an exercise is in it: show example of complete metric space and a sequence of embedded closed balls, that have empty intersection". I understand, that sequence of their diameters must not converge, but I could not find the example. I was thinking about space of infinite sequences $(x_1,x_2, x_3, \dots): |x_i|<=1$with distance between x and y $\rho(x,y)=max(|x_i-y_i|)_i$, and sequence of balls with radius 1 and centers $x_0=(0,0,\dots), x_1=(1,0,0,\dots), x_2=(1,1,0,\dots),etc$. But is it correct?
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What counts as a sequence? Do the balls have to be nested? – Cheerful Parsnip May 15 '13 at 19:11
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1The first answer here will help. – David Mitra May 15 '13 at 19:12
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@DavidMitra, yes, this is the answer, I guess, thanks a lot. – aptypr May 16 '13 at 02:08