Proof of Convolution with a $C^k$ Kernel is of Class $C^k$

Claim 1. We argue by induction on $k$, the assertion being proved simultaneously for all data $n$, $\Omega$, $f$, $\delta$, $\rho$ admissible in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel. Throughout, $d$ is the Euclidean distance, and by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions (taken with $E=\Omega^{\delta}$) a real-valued function on $\Omega^{\delta}$ is continuous at every point of $\Omega^{\delta}$ exactly when it is continuous on $\Omega^{\delta}$ as a map into $(\mathbb{R},d)$. Recall also that the scalar convention of clause 3 of C^k Maps on a Euclidean Open Set reads a real-valued function as a map into $\mathbb{R}^1$ with itself as single coordinate function.

Base case $k=1$. Suppose $\rho$ is of class $C^1$ on $\mathbb{R}^n$. By The Convolution of a Continuous Function with a Continuous Kernel is Continuous, $f*\rho$ is continuous on $\Omega^{\delta}$ as a map into $(\mathbb{R},d)$, hence continuous at every point of $\Omega^{\delta}$. Let $i\in\{1,\dots,n\}$. By claim 2 of Differentiating a Convolution through the Kernel, the partial derivative of $f*\rho$ with respect to the $i$th variable exists at every point of $\Omega^{\delta}$ and the resulting function is $f*(\partial_i\rho)$. By claim 1 of the same lemma, $\partial_i\rho$ is again an admissible kernel in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel for the same $n$, $\Omega$, $f$ and $\delta$, so The Convolution of a Continuous Function with a Continuous Kernel is Continuous applies to it and shows that $f*(\partial_i\rho)$ is continuous on $\Omega^{\delta}$, hence continuous at every point of $\Omega^{\delta}$. As $i$ was arbitrary, clauses 1 and 3 of C^k Maps on a Euclidean Open Set give that $f*\rho$ is of class $C^1$ on $\Omega^{\delta}$.

Induction step. Let $k$ be a natural number and assume the assertion for $k$, for all admissible data. Suppose $\rho$ is of class $C^{k+1}$ on $\mathbb{R}^n$. By clauses 2 and 3 of C^k Maps on a Euclidean Open Set, $\rho$ is of class $C^1$ on $\mathbb{R}^n$ and, for every $i\in\{1,\dots,n\}$, the function $\partial_i\rho$ is of class $C^k$ on $\mathbb{R}^n$. By the base case, $f*\rho$ is of class $C^1$ on $\Omega^{\delta}$, and by claim 2 of Differentiating a Convolution through the Kernel its partial derivative with respect to the $i$th variable is the function $f*(\partial_i\rho)$ on $\Omega^{\delta}$. By claim 1 of that lemma, $\partial_i\rho$ is an admissible kernel for the same $n$, $\Omega$, $f$ and $\delta$, so the induction hypothesis applied to the data $n$, $\Omega$, $f$, $\delta$, $\partial_i\rho$ shows that $f*(\partial_i\rho)$ is of class $C^k$ on $\Omega^{\delta}$. Since this holds for every $i$, clauses 2 and 3 of C^k Maps on a Euclidean Open Set give that $f*\rho$ is of class $C^{k+1}$ on $\Omega^{\delta}$.

Claim 2. Suppose $\rho$ is smooth on $\mathbb{R}^n$. By Smooth Map on a Euclidean Open Set, $\rho$ is of class $C^k$ on $\mathbb{R}^n$ for every natural number $k$, so claim 1 gives that $f*\rho$ is of class $C^k$ on $\Omega^{\delta}$ for every natural number $k$; by Smooth Map on a Euclidean Open Set again, $f*\rho$ is smooth on $\Omega^{\delta}$. $\blacksquare$