= Solution
The <Rao-Blackwell theorem> states that if $T$ is sufficient and $U$ is an estimator with finite variance, then
$$
U^*=\mathbb E[U\mid T]
$$
has the same expectation as $U$ and no larger variance; it preserves unbiasedness. The <tower property of conditional expectation> gives
$$
\mathbb E U^*=\mathbb E U.
$$
The <law of total variance> gives
$$
\operatorname{Var}(U)
=\operatorname{Var}(\mathbb E[U\mid T])
+\mathbb E[\operatorname{Var}(U\mid T)]
\geq\operatorname{Var}(U^*).
$$
Sufficiency ensures that $U^*$ is a statistic whose definition does not depend on the unknown parameter. The inequality is strict exactly when the conditional variance is positive with positive probability.
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