Only sort of true. It doesn't make sense to compare n dimensional volume to n+1 dimensional volumes, so the limit of the volume of an n-sphere isn't meaningful. The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.
In particular, it's equally valid and frankly nicer to define the unit n-sphere to be volume 1 rather than the unit cube. Do that and we see that this statement is just saying that the n-cube grows in volume to infinity, which makes sense given the fact you point out that it contains points increasingly far from the origin.
I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
> The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.
This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because...
> I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat!
Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.
In particular, it's equally valid and frankly nicer to define the unit n-sphere to be volume 1 rather than the unit cube. Do that and we see that this statement is just saying that the n-cube grows in volume to infinity, which makes sense given the fact you point out that it contains points increasingly far from the origin.
I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.