Prove that

**Solution**

Let be independent and uniform random variables. By the law of large numbers we have

in probability as . Therefore ,

in probability as .

The ratio random variables are bounded below by zero and above by one. This guarantees convergence of the expectations, as well.

So,

which is the required result.

**Remark: **In general it holds that

because the distribution of is the Lebesgue measure on hence for every measurable function ,