A metric space, (M,d), is bounded if there exists an r>0 such that d(x,y) ≤ r for all x and y in M, and totally bounded if there for every r>0 exists a finite number of 'open balls' with radius r whose union totally covers M.

So far so good and it's not hard to see why a totally bounded metric space is also bounded. However I have some trouble seeing why the reverse - why a bounded space isn't also totally bounded, is true and I suppose my question boils down to:

Is it because an infinite metric space, (

**R**,|x-y|) say, is bounded by r=∞ but you can't cover the infinite set with a finite number of open balls (unless the distance metric is something like d(x,y)={∞ if x≠y or 0 otherwise} then it is also totally bounded? but that is hardly a useful metric I suppose and not very 'general')? Does that mean then that all metric spaces are bounded?

I'm not sure that is making sense - at all.

Thank you.

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