"Proved" is a strong word.

I might suggest "investigated" (as in investigated "the problem of the extent to which the three-dimensional nature of physical space is determined by the structure of basic physical equations or is reflected in these basic equations"), "postulated", or even "demonstrated" (as in postulated or demonstrated "that the stability requirement

for n–dimensional two–body planetary system very strongly constrains space dimensionality, favouring 3–d").

In any case, Ehrenfest's work, like all other scientific work, is based to some degree on assumptions and other work that may be found to be incomplete or invalid in a given context that is as yet not completely or perfectly understood.

https://mathshistory.st-andrews.ac.uk/Biographies/Ehrenfest/

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1056.9964&rep=rep1&type=pdf