Let be i.i.d. random variables uniformly distributed over . Since ,
with probability one. Moreover, by the strong law of large numbers (SLLN),
holds with probability one. So by the dominated convergence theorem,
继续推广：
With , , , and
另一个碎片：
Since is equidistributed modulo , the limit could be rewritten as the limit of the expected value of the geometric average of uniform random variables. The integral for this would be
This can actually be rewritten as
since each is independent of the others. The inner integral is then equal to , so the limit is
