Complete metric space

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mario22

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Dec 11, 2010
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Hi
please read the following question:

Let N = {1, 2, ...}. For all x, y belong to N set d(x,y)=|x-y|(xy)[sup:14if34ks]-1[/sup:14if34ks]
Show that d is a metric and verify if the metric space (X, d) is complete.

I already showed that d is a metric but I couldn't prove that the metric space is complete.
Can anyone give me a counter example - maybe it is not a complete metric space?

Thanks,
Mario
 
how come 0 is a limit point? there is no subset that converge to 0.
pay attention that N = {1, 2, ...} - natural numbers.

Thanks,
Mario
 
I think you're confused because you're thinking that the distance between 2 consecutive natural numbers is 1. This is the case when we usually think about natural numbers, but it is not the case using the metric as defined in this problem.

So, for example, the distance between 1 and 2 is 1/2 in this metric. The distance between 2 and 3 is 1/6. The distance between 3 and 4 is 1/12. In general, the distance between n and n+1 is 1/[n(n+1)] which goes to 0 as n goes to infinity.
 
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