[proofplan]
First approximate $f$ uniformly on $\overline D$ by a function holomorphic on an open neighbourhood of $\overline D$; this is the Mergelyan approximation theorem for smoothly bounded strongly pseudoconvex domains, whose analytic input is solvability of the antiholomorphic Cauchy-Riemann equation on strongly pseudoconvex collars. Then use the hypothesis that $\overline D$ is $\mathcal O(\Omega)$-convex to apply the Oka-Weil approximation theorem inside the pseudoconvex domain $\Omega$. Splitting the tolerance into two halves gives the required uniform estimate for the final [holomorphic function](/page/Holomorphic%20Function) on $\Omega$.
[/proofplan]
[step:Approximate $f$ by a function holomorphic near $\overline D$]
Let $K := \overline D$, viewed as a compact subset of $\Omega$. Define the tolerance
\begin{align*}
\delta := \frac{\varepsilon}{2}.
\end{align*}
Since $D \Subset \Omega$ is smoothly bounded and strongly pseudoconvex, and since $f \in C(K) \cap \mathcal O(D)$, the local Mergelyan approximation theorem for smoothly bounded strongly pseudoconvex domains applies to the pair $(D,f)$ with tolerance $\delta$. Hence there exist an [open set](/page/Open%20Set) $U \subset \mathbb C^n$ with $K \subset U$ and a function
\begin{align*}
g: U &\to \mathbb C
\end{align*}
holomorphic on $U$ such that
\begin{align*}
\sup_{z \in K} |f(z)-g(z)| < \delta.
\end{align*}
[guided]
The first task is to remove the [boundary regularity](/theorems/99) issue. The function $f$ is only assumed continuous on $K = \overline D$ and holomorphic on the interior $D$, so it is not initially eligible for Oka-Weil approximation, which requires a function holomorphic on a neighbourhood of the compact set being approximated.
We use the local Mergelyan approximation theorem for smoothly bounded strongly pseudoconvex domains. Its hypotheses are exactly the local hypotheses available here: $D \Subset \Omega$ is bounded with smooth strongly pseudoconvex boundary, and $f \in C(\overline D) \cap \mathcal O(D)$. Applying that theorem with the positive tolerance
\begin{align*}
\delta := \frac{\varepsilon}{2}
\end{align*}
gives an open neighbourhood $U \subset \mathbb C^n$ of $K := \overline D$ and a holomorphic map
\begin{align*}
g: U &\to \mathbb C
\end{align*}
such that
\begin{align*}
\sup_{z \in K} |f(z)-g(z)| < \delta.
\end{align*}
This is the point where smooth strong pseudoconvexity of $D$ is used: it is the geometric hypothesis that supplies the local Mergelyan approximation across the boundary.
[/guided]
[/step]
[step:Approximate the neighbourhood-holomorphic function by functions in $\mathcal O(\Omega)$]
The compact set $K=\overline D$ is $\mathcal O(\Omega)$-convex by hypothesis. Since $\Omega \subset \mathbb C^n$ is pseudoconvex, the [Solution of the Levi Problem](/theorems/3416) implies that $\Omega$ is a Stein domain, so the [Oka-Weil Approximation Theorem](/theorems/3415) applies on $\Omega$ to the $\mathcal O(\Omega)$-convex compact set $K$ and to the function $g$, which is holomorphic on the open neighbourhood $U$ of $K$. Therefore there exists a function
\begin{align*}
F: \Omega &\to \mathbb C
\end{align*}
holomorphic on $\Omega$ such that
\begin{align*}
\sup_{z \in K} |g(z)-F(z)| < \delta.
\end{align*}
[guided]
Now the function $g$ has the correct form for Oka-Weil approximation: it is holomorphic on an open neighbourhood $U$ of the compact set $K = \overline D$. We must verify the ambient and convexity hypotheses before applying the theorem.
First, $K$ is $\mathcal O(\Omega)$-convex by the theorem statement. Second, $\Omega \subset \mathbb C^n$ is pseudoconvex, so by the [Solution of the Levi Problem](/theorems/3416) the domain $\Omega$ is Stein. Thus the [Oka-Weil Approximation Theorem](/theorems/3415) applies to the Stein domain $\Omega$, the $\mathcal O(\Omega)$-convex compact set $K$, and the neighbourhood-[holomorphic function](/page/Holomorphic%20Function)
\begin{align*}
g: U &\to \mathbb C.
\end{align*}
With the same tolerance
\begin{align*}
\delta = \frac{\varepsilon}{2},
\end{align*}
Oka-Weil gives a holomorphic map
\begin{align*}
F: \Omega &\to \mathbb C
\end{align*}
satisfying
\begin{align*}
\sup_{z \in K} |g(z)-F(z)| < \delta.
\end{align*}
This is where the $\mathcal O(\Omega)$-convexity assumption is used: without it, Oka-Weil approximation by global holomorphic functions on $\Omega$ is not guaranteed.
[/guided]
[/step]
[step:Combine the two uniform estimates on $\overline D$]
For every $z \in K$, the triangle inequality in $\mathbb C$ gives
\begin{align*}
|f(z)-F(z)| \le |f(z)-g(z)| + |g(z)-F(z)|.
\end{align*}
Taking the supremum over $z \in K$ and using the two preceding estimates yields
\begin{align*}
\sup_{z \in \overline D} |f(z)-F(z)|
&= \sup_{z \in K} |f(z)-F(z)| \\
&\le \sup_{z \in K} |f(z)-g(z)| + \sup_{z \in K} |g(z)-F(z)| \\
&< \delta + \delta \\
&= \varepsilon.
\end{align*}
Since $F \in \mathcal O(\Omega)$, this is the required approximating function.
[/step]
