= Solution
Write $u(y,\theta)=\partial_\theta\log f(y,\theta)$ for the single-observation <score function>. Differentiating the normalization of the <probability density function> gives
$$
\mathbb E_\theta[u(Y,\theta)]
=\int f(y,\theta)\partial_\theta\log f(y,\theta)\,dy
=\partial_\theta\int f(y,\theta)\,dy=0.
$$
For discrete observations, use a sum instead. The allowed interchange of differentiation and integration is the usual regular-model assumption with common support. Adding these identities gives $\mathbb E_\theta U_n=0$.
Let $p_\theta$ be the joint density. Because $T$ is an <unbiased estimator> with no explicit dependence on the parameter, differentiating $\mathbb E_\theta T=\theta$ gives
$$
1=\int T\,\partial_\theta p_\theta
=\mathbb E_\theta[TU_n]
=\operatorname{Cov}_\theta(T,U_n).
$$
The scores are <independent random variables> with mean zero, so <variance additivity for independent random variables> gives $\operatorname{Var}_\theta(U_n)=n\operatorname{Var}_\theta(U_1)$. The <Cauchy-Schwarz inequality> applied to the centred estimator and the score now gives the <Cramér-Rao lower bound>:
$$
\boxed{\operatorname{Var}_\theta(T)\geq\frac1{n\operatorname{Var}_\theta(U_1)}.}
$$
The denominator is the sample <Fisher information>. This bound concerns regular models with positive finite information; the preceding covariance identity explains why a small estimator variance requires a sufficiently informative score.
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