This reasoning is pretty weird, but the conclusion is basically right. That is, there is absolutely no way to extend the conventional notion of volume to Rinfinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space. Edit: what I mean by Rinfinity is a bit ambiguous, but let's say for the purpose of a hypersphere we want something like l^2 hilbert space to ensure no vectors with infinite length appear, then we have a separable space and the proof is complete.