= Solution
The risk of $\overline X_n$ is constantly $1/n$. Suppose an estimator $\delta$ dominates it, and put
$m(\theta)=\mathbb E_\theta\delta$ and $b(\theta)=m(\theta)-\theta$. The biased <Cramér-Rao bound>, using total information $n$, gives
$$
R(\theta,\delta)\geq
\frac{m'(\theta)^2}{n}+b(\theta)^2.
$$
Domination would imply
$$
(1+b'(\theta))^2+n b(\theta)^2\leq1
$$
for every real $\theta$. The standard differential-inequality argument shows that the only globally defined differentiable solution is $b\equiv0$: a nonzero value forces $b'$ to retain a sign and magnitude that makes $b$ leave the permitted bounded interval in one time direction.
Thus $m(\theta)=\theta$. The Cramer--Rao inequality now gives
$\operatorname{Var}_\theta\delta\geq1/n$, so domination forces equality everywhere. Equality in Cramer--Rao makes the centred estimator proportional to the Gaussian score:
$$
\delta-\theta=\frac1n\sum_i(X_i-\theta),
$$
and hence $\delta=\overline X_n$ almost surely. No strict improvement exists, so $\overline X_n$ is admissible.
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