[proofplan]
The proof is the standard inversion of Hotelling's one-sample $T^2$ statistic. First we record the null distribution of the statistic obtained by evaluating the quadratic form at the true mean $\mu$. Then we rewrite the event $\{\mu \in \mathcal{E}_\alpha\}$ as exactly the event that this statistic falls below its $(1-\alpha)$ critical value. The quantile identity for the continuous $F_{p,n-p}$ distribution gives the desired coverage probability.
[/proofplan]
[step:Identify the Hotelling statistic at the true mean]
Define the Hotelling statistic $T^2:\Omega\to[0,\infty)$ by
\begin{align*}
T^2
=
n(\bar X-\mu)^\top S^{-1}(\bar X-\mu),
\end{align*}
on the almost sure event on which $S$ is invertible. Since $n>p$ and $\Sigma$ is positive definite, the sample covariance matrix $S$ is positive definite almost surely under the multivariate normal model, so this definition is valid almost surely.
By Hotelling's one-sample null distribution theorem, applied to the independent sample $X_1,\dots,X_n\overset{\mathrm{i.i.d.}}{\sim}\mathcal{N}_p(\mu,\Sigma)$ with $n>p$ and $\Sigma$ positive definite,
\begin{align*}
\frac{n-p}{p(n-1)}T^2 \sim F_{p,n-p}.
\end{align*}
(citing a result not yet in the wiki: Hotelling's one-sample $T^2$ null distribution)
[guided]
We need the distribution of the quadratic form that appears in the proposed confidence ellipsoid when the candidate center $m$ is equal to the true mean $\mu$. Define the random variable
\begin{align*}
T^2:\Omega &\to [0,\infty)\\
\omega &\mapsto n(\bar X(\omega)-\mu)^\top S(\omega)^{-1}(\bar X(\omega)-\mu),
\end{align*}
on the event where $S(\omega)$ is invertible. The inverse is legitimate almost surely: under the multivariate normal model with positive definite covariance matrix $\Sigma$, the scaled sample covariance matrix $(n-1)S$ has a nonsingular Wishart distribution when its degrees of freedom satisfy $n-1\ge p$, and this is guaranteed by $n>p$.
The distributional input is Hotelling's one-sample $T^2$ null distribution. It says that for independent random vectors $X_1,\dots,X_n\overset{\mathrm{i.i.d.}}{\sim}\mathcal{N}_p(\mu,\Sigma)$, with $\Sigma$ positive definite and $n>p$, the scaled statistic satisfies
\begin{align*}
\frac{n-p}{p(n-1)}T^2 \sim F_{p,n-p}.
\end{align*}
This is precisely the null distribution needed for the confidence set, because a confidence region for $\mu$ is obtained by retaining exactly those candidate means $m$ for which the corresponding Hotelling statistic is not too large. Here we only need the statistic at the true value $m=\mu$.
(citing a result not yet in the wiki: Hotelling's one-sample $T^2$ null distribution)
[/guided]
[/step]
[step:Rewrite membership of the true mean as a Hotelling critical event]
By the definition of $\mathcal{E}_\alpha$, the event that the true mean belongs to the random ellipsoid is
\begin{align*}
\{\mu \in \mathcal{E}_\alpha\}
=
\left\{
n(\bar X-\mu)^\top S^{-1}(\bar X-\mu)
\le
\frac{p(n-1)}{n-p}F_{p,n-p;1-\alpha}
\right\}.
\end{align*}
Using the definition of $T^2$, this is equivalently
\begin{align*}
\{\mu \in \mathcal{E}_\alpha\}
=
\left\{
\frac{n-p}{p(n-1)}T^2
\le
F_{p,n-p;1-\alpha}
\right\}.
\end{align*}
[/step]
[step:Apply the $F$ quantile identity to compute the coverage]
Let $Y:\Omega\to[0,\infty)$ be the random variable
\begin{align*}
Y=\frac{n-p}{p(n-1)}T^2.
\end{align*}
From the previous distributional step, $Y\sim F_{p,n-p}$. Since the $F_{p,n-p}$ distribution is continuous and $F_{p,n-p;1-\alpha}$ is its $(1-\alpha)$-quantile,
\begin{align*}
\mathbb{P}_\mu\left(Y\le F_{p,n-p;1-\alpha}\right)=1-\alpha.
\end{align*}
Substituting the event identified in the preceding step gives
\begin{align*}
\mathbb{P}_\mu(\mu\in\mathcal{E}_\alpha)
=
\mathbb{P}_\mu\left(
\frac{n-p}{p(n-1)}T^2
\le
F_{p,n-p;1-\alpha}
\right)
=
1-\alpha.
\end{align*}
Thus $\mathcal{E}_\alpha$ has exact coverage probability $1-\alpha$ for $\mu$, so it is a $(1-\alpha)$ confidence region.
[/step]
