Solution
= Solution
Because $U$ and $V$ are bounded, their exponential series may be integrated term by term for every real $t$. Thus their <moment-generating functions> satisfy
$$
M_U(t)=\mathbb E[e^{tU}]
=\sum_{j=0}^{\infty}\frac{t^j}{j!}\mathbb E[U^j],
$$
and similarly for $V$. Equality of every moment gives equality term by term, so
$$
\boxed{M_U(t)=M_V(t)\quad\text{for every }t\in\mathbb R}.
$$