[proofplan]
We approximate the full index set $F$ by finite $d_P$-nets. On each finite net, the assumed finite-dimensional convergence gives [weak convergence](/page/Weak%20Convergence) of the corresponding projected empirical processes. Asymptotic equicontinuity makes the empirical process uniformly close to its finite-net projection, while the assumed uniform $d_P$-continuity of $G_P$ gives the analogous approximation for the Gaussian limit. Testing against bounded Lipschitz functions on $\ell^\infty(F)$ and then sending first $n\to\infty$ and then the net mesh to $0$ transfers the finite-dimensional convergence to weak convergence in $\ell^\infty(F)$.
[/proofplan]
[step:Choose finite nets and define projection operators]
Fix $\delta>0$. Since $(F,d_P)$ is [totally bounded](/page/Totally%20Bounded), there exist $m_\delta\in\mathbb N$ and functions $f_{\delta,1},\dots,f_{\delta,m_\delta}\in F$ such that
\begin{align*}
F\subset \bigcup_{j=1}^{m_\delta}\{f\in F:d_P(f,f_{\delta,j})<\delta\}.
\end{align*}
Define a map $\pi_\delta:F\to \{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$ by choosing, for each $f\in F$, the least index $j$ such that $d_P(f,f_{\delta,j})<\delta$, and setting $\pi_\delta(f):=f_{\delta,j}$. Then
\begin{align*}
d_P(f,\pi_\delta(f))<\delta
\end{align*}
for every $f\in F$.
Define the finite-coordinate restriction map
\begin{align*}
R_\delta:\ell^\infty(F)\to\mathbb R^{m_\delta}, \qquad z\mapsto (z(f_{\delta,1}),\dots,z(f_{\delta,m_\delta})).
\end{align*}
and define the extension map
\begin{align*}
E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F), \qquad a\mapsto \bigl(f\mapsto a_j\text{ where }\pi_\delta(f)=f_{\delta,j}\bigr).
\end{align*}
The map $E_\delta$ is continuous when $\mathbb R^{m_\delta}$ is equipped with the maximum norm and $\ell^\infty(F)$ is equipped with the supremum norm, because
\begin{align*}
\|E_\delta a-E_\delta b\|_{\ell^\infty(F)}\le \max_{1\le j\le m_\delta}|a_j-b_j|
\end{align*}
for all $a,b\in\mathbb R^{m_\delta}$.
For each $n\in\mathbb N$, define the projected empirical process
\begin{align*}
G_{n,\delta}:=E_\delta R_\delta G_n.
\end{align*}
Define likewise
\begin{align*}
G_{P,\delta}:=E_\delta R_\delta G_P.
\end{align*}
Thus, for every $f\in F$,
\begin{align*}
G_{n,\delta}(f)=G_n(\pi_\delta(f))
\end{align*}
and
\begin{align*}
G_{P,\delta}(f)=G_P(\pi_\delta(f)).
\end{align*}
[guided]
The role of [total boundedness](/page/Total%20Boundedness) is to reduce the potentially infinite index set $F$ to finitely many representatives. Fix $\delta>0$. Total boundedness of $(F,d_P)$ says precisely that finitely many $d_P$-balls of radius $\delta$ cover $F$. Hence we may choose $m_\delta\in\mathbb N$ and functions $f_{\delta,1},\dots,f_{\delta,m_\delta}\in F$ such that
\begin{align*}
F\subset \bigcup_{j=1}^{m_\delta}\{f\in F:d_P(f,f_{\delta,j})<\delta\}.
\end{align*}
For each $f\in F$, at least one of these centres lies within $d_P$-distance $\delta$ of $f$. To make a genuine function rather than a multivalued choice, define $\pi_\delta:F\to \{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$ by choosing the least index $j$ such that $d_P(f,f_{\delta,j})<\delta$, and set $\pi_\delta(f):=f_{\delta,j}$. This finite tie-breaking rule gives a well-defined map and ensures
\begin{align*}
d_P(f,\pi_\delta(f))<\delta
\end{align*}
for every $f\in F$.
Now we separate two operations. First, restrict a [bounded function](/page/Bounded%20Function) on $F$ to the finite net by defining
\begin{align*}
R_\delta:\ell^\infty(F)\to\mathbb R^{m_\delta}, \qquad z\mapsto (z(f_{\delta,1}),\dots,z(f_{\delta,m_\delta})).
\end{align*}
Second, extend a vector of net values back to all of $F$ by making it constant on the cells determined by $\pi_\delta$:
\begin{align*}
E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F), \qquad a\mapsto \bigl(f\mapsto a_j\text{ where }\pi_\delta(f)=f_{\delta,j}\bigr).
\end{align*}
This extension is continuous. Indeed, if $a,b\in\mathbb R^{m_\delta}$, then for each $f\in F$ the values $(E_\delta a)(f)$ and $(E_\delta b)(f)$ are two coordinates with the same index $j$, so
\begin{align*}
|(E_\delta a)(f)-(E_\delta b)(f)|\le \max_{1\le j\le m_\delta}|a_j-b_j|.
\end{align*}
Taking the supremum over $f\in F$ gives
\begin{align*}
\|E_\delta a-E_\delta b\|_{\ell^\infty(F)}\le \max_{1\le j\le m_\delta}|a_j-b_j|.
\end{align*}
We then define
\begin{align*}
G_{n,\delta}:=E_\delta R_\delta G_n
\end{align*}
and
\begin{align*}
G_{P,\delta}:=E_\delta R_\delta G_P.
\end{align*}
These are the finite-net approximations to $G_n$ and $G_P$. Pointwise on $F$, they satisfy
\begin{align*}
G_{n,\delta}(f)=G_n(\pi_\delta(f))
\end{align*}
and
\begin{align*}
G_{P,\delta}(f)=G_P(\pi_\delta(f)).
\end{align*}
[/guided]
[/step]
[step:Obtain weak convergence on each finite net]
By the finite-dimensional convergence hypothesis applied to the finite set $\{f_{\delta,1},\dots,f_{\delta,m_\delta}\}$,
\begin{align*}
R_\delta G_n\xrightarrow{d}R_\delta G_P
\end{align*}
in $\mathbb R^{m_\delta}$. The vector $R_\delta G_n$ is an ordinary finite-dimensional random vector in the measurable case; in the outer-probability case this finite-dimensional convergence is part of the hypothesis and is interpreted in the usual Borel sense on $\mathbb R^{m_\delta}$. Since $E_\delta:\mathbb R^{m_\delta}\to\ell^\infty(F)$ is continuous, the [continuous mapping theorem](/theorems/1847), or equivalently the bounded-Lipschitz form of its outer-probability version, gives
\begin{align*}
G_{n,\delta}=E_\delta R_\delta G_n\xrightarrow{d}E_\delta R_\delta G_P=G_{P,\delta}
\end{align*}
in $\ell^\infty(F)$ for each fixed $\delta>0$.
[/step]
[step:Control the empirical projection error by asymptotic equicontinuity]
For each $n\in\mathbb N$ and $\delta>0$,
\begin{align*}
\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}
=\sup_{f\in F}|G_n(f)-G_n(\pi_\delta(f))|.
\end{align*}
Since $d_P(f,\pi_\delta(f))<\delta$ for every $f\in F$,
\begin{align*}
\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}
\le
\sup_{\substack{f, g\in F, d_P(f, g)<\delta}}|G_n(f)-G_n(g)|.
\end{align*}
Therefore, for every $\eta>0$,
\begin{align*}
\lim_{\delta\downarrow0}\limsup_{n\to\infty}\mathbb P^*\left(\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}>\eta\right)=0
\end{align*}
by asymptotic equicontinuity.
[/step]
[step:Control the Gaussian projection error by uniform continuity]
Because $G_P$ has bounded uniformly $d_P$-continuous sample paths on the [probability space](/page/Probability%20Space) $(\Omega_P,\mathcal A_P,\mathbb P_P)$ from the theorem statement, for $\mathbb P_P$-almost every $\omega\in\Omega_P$ the map $G_P(\omega):F\to\mathbb R$, defined by $f\mapsto G_P(f)(\omega)$, is bounded and uniformly continuous with respect to $d_P$. Hence, for such $\omega$,
\begin{align*}
\|G_P(\omega)-G_{P,\delta}(\omega)\|_{\ell^\infty(F)}
=
\sup_{f\in F}|G_P(f)(\omega)-G_P(\pi_\delta(f))(\omega)|
\end{align*}
is bounded above by
\begin{align*}
\sup_{\substack{f, g\in F, d_P(f, g)<\delta}}|G_P(f)(\omega)-G_P(g)(\omega)|.
\end{align*}
The right-hand side tends to $0$ as $\delta\downarrow0$ by uniform $d_P$-continuity of the sample path. Thus
\begin{align*}
\|G_P-G_{P,\delta}\|_{\ell^\infty(F)}\xrightarrow{a.s.}0
\end{align*}
as $\delta\downarrow0$.
[/step]
[step:Transfer convergence from finite nets to $\ell^\infty(F)$]
Let $\varphi:\ell^\infty(F)\to\mathbb R$ be a bounded [Lipschitz function](/page/Lipschitz%20Function). Define
\begin{align*}
\|\varphi\|_\infty:=\sup_{z\in\ell^\infty(F)}|\varphi(z)|
\end{align*}
and let $L_\varphi\ge0$ be a Lipschitz constant for $\varphi$ with respect to the supremum norm on $\ell^\infty(F)$.
Fix $\varepsilon>0$. By the Gaussian projection estimate, choose $\delta>0$ so small that
\begin{align*}
\mathbb E\left[\min\{1,\|G_P-G_{P,\delta}\|_{\ell^\infty(F)}\}\right]<\varepsilon.
\end{align*}
By the empirical projection estimate, after possibly decreasing $\delta$, we also have
\begin{align*}
\limsup_{n\to\infty}\mathbb P^*\left(\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}>\varepsilon\right)<\varepsilon.
\end{align*}
Using the Lipschitz property and boundedness of $\varphi$, we obtain
\begin{align*}
|\varphi(G_n)-\varphi(G_{n,\delta})|
\le
L_\varphi\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}
\end{align*}
on the event $\{\|G_n-G_{n,\delta}\|_{\ell^\infty(F)}\le\varepsilon\}$, while the absolute difference is at most $2\|\varphi\|_\infty$ everywhere. Hence
\begin{align*}
\limsup_{n\to\infty}\left|\mathbb E[\varphi(G_n)]-\mathbb E[\varphi(G_{n,\delta})]\right|
\le
L_\varphi\varepsilon+2\|\varphi\|_\infty\varepsilon.
\end{align*}
Similarly,
\begin{align*}
\left|\mathbb E[\varphi(G_P)]-\mathbb E[\varphi(G_{P,\delta})]\right|
\le
L_\varphi\mathbb E\left[\min\{\|G_P-G_{P,\delta}\|_{\ell^\infty(F)},1\}\right]+2\|\varphi\|_\infty\mathbb P(\|G_P-G_{P,\delta}\|_{\ell^\infty(F)}>1),
\end{align*}
and the right-hand side tends to $0$ as $\delta\downarrow0$.
For the fixed $\delta$ chosen above, the finite-net convergence gives
\begin{align*}
\lim_{n\to\infty}\mathbb E[\varphi(G_{n,\delta})]
=
\mathbb E[\varphi(G_{P,\delta})].
\end{align*}
Combining the three estimates gives
\begin{align*}
\limsup_{n\to\infty}\left|\mathbb E[\varphi(G_n)]-\mathbb E[\varphi(G_P)]\right|
\le
L_\varphi\varepsilon+2\|\varphi\|_\infty\varepsilon+o_\delta(1),
\end{align*}
where $o_\delta(1)$ denotes a quantity that tends to $0$ as $\delta\downarrow0$. Since $\varepsilon>0$ was arbitrary, this proves
\begin{align*}
\lim_{n\to\infty}\mathbb E[\varphi(G_n)]
=
\mathbb E[\varphi(G_P)]
\end{align*}
for every bounded Lipschitz $\varphi:\ell^\infty(F)\to\mathbb R$.
In the Borel measurable case, the bounded-Lipschitz characterization of weak convergence on metric spaces applies to the Borel laws of $G_n$ and $G_P$. Therefore
\begin{align*}
G_n\xrightarrow{d}G_P
\end{align*}
in $\ell^\infty(F)$.
In the outer-probability case, replace every expectation involving $G_n$ in the preceding bounded-Lipschitz estimate by outer expectation $\mathbb E^*$ and every probability involving $G_n$ by outer probability $\mathbb P^*$. The projection estimate for $G_n$ was already proved with $\mathbb P^*$, and the finite-net convergence term is ordinary because it concerns the finite-dimensional vector $R_\delta G_n$ and the continuous finite-net image $E_\delta R_\delta G_n$. Asymptotic measurability ensures that the bounded Lipschitz functions $\varphi(G_n)$ have outer and inner expectations asymptotically equal, so the preceding estimate gives
\begin{align*}
\mathbb E^*\varphi(G_n)\to\mathbb E\varphi(G_P)
\end{align*}
for every bounded Lipschitz $\varphi:\ell^\infty(F)\to\mathbb R$. By the bounded-Lipschitz outer-expectation criterion stated in the theorem, this is precisely
\begin{align*}
G_n\xrightarrow{d}G_P
\end{align*}
in $\ell^\infty(F)$ under the standard empirical-process convention. This completes the proof.
[/step]
