= Solution
Let
$$
R=\begin{pmatrix}I_k&0\end{pmatrix}\in\mathbb R^{k\times p},
\qquad
r=R\theta^*.
$$
Then the null hypothesis is $R\theta_0=r$. Define the <Wald statistic for linear restrictions>
$$
W_{n,R}
=n(R\widehat\theta_n-r)^T
\left[R I(\widehat\theta_n)^{-1}R^T\right]^{-1}
(R\widehat\theta_n-r).
$$
Under $H_0$, part (a) and the continuous mapping theorem give
$$
\sqrt n(R\widehat\theta_n-r)
=\sqrt nR(\widehat\theta_n-\theta_0)
\xrightarrow d N_k(0,V_0),
$$
where
$$
V_0=R I(\theta_0)^{-1}R^T.
$$
The matrix $V_0$ is positive definite because $I(\theta_0)$ is positive definite and $R$ has full row rank. Consistency gives
$$
R I(\widehat\theta_n)^{-1}R^T\xrightarrow p V_0.
$$
Therefore <Slutsky theorem> yields
$$
V_0^{-1/2}\sqrt n(R\widehat\theta_n-r)
\xrightarrow dN_k(0,I_k),
$$
and applying the continuous squared-norm map proves rigorously that
$$
\boxed{W_{n,R}\xrightarrow d\chi_k^2.}
$$
Let $q_{k,1-\alpha}$ be the $(1-\alpha)$ quantile of $\chi_k^2$. Rejecting $H_0$ when
$$
W_{n,R}>q_{k,1-\alpha}
$$
has rejection probability tending to $\alpha$ under every fixed parameter satisfying $H_0$, and is therefore an asymptotically valid level-$\alpha$ test.
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