= Solution
For <scalar> $\theta$, Taylor's theorem about the MLE, where
$\ell_n'(\widehat\theta_n)=0$, gives some $\widetilde\theta_n$ between $\theta_0$ and $\widehat\theta_n$ such that
$$
\Lambda_n
=-\ell_n''(\widetilde\theta_n)
(\widehat\theta_n-\theta_0)^2.
$$
The Wald statistic is
$$
W_n(\theta_0)
=nI(\widehat\theta_n)(\widehat\theta_n-\theta_0)^2.
$$
Under the null, consistency gives
$\widehat\theta_n,\widetilde\theta_n\to\theta_0$ in probability. A uniform law of large numbers for the observed information on a neighborhood of $\theta_0$ gives
$$
-\frac1n\ell_n''(\widetilde\theta_n)\xrightarrow P I(\theta_0),
\qquad
I(\widehat\theta_n)\xrightarrow P I(\theta_0)>0.
$$
Cancellation of the common squared displacement and Slutsky's theorem prove the <Wald and likelihood-ratio asymptotic equivalence>
$$
\boxed{\frac{\Lambda_n}{W_n(\theta_0)}\xrightarrow P1}.
$$
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