 Solution

The inverse transformation of a random variable is $u=e^{-v}$, with absolute derivative $e^{-v}$. The condition $0<u<b$ becomes $v>-\log b$. The change of variables for probability densities givesEquivalently, $V_n+\log b$ has the exponential distribution of rate one. Applying a separate measurable transformation of a random variable to each independent $U_n$ preserves independence of random variables, so the $V_n$ are independent and identically distributed.