A diagonal subsequence of approximations of approximations converges in $C^\infty_{\mathrm{loc}}$

Let $E$ be a proper real normed space (e.g. $\mathbb{R}^n$) and $F$ a real normed space. Let $f:E\to F$, let $f_k:E\to F$ ($k\in\mathbb{N}$) satisfy $f_k\to f$ in $C^\infty_{\mathrm{loc}}$, and for each $k$ let $g_{k,j}:E\to F$ ($j\in\mathbb{N}$) satisfy $g_{k,j}\to f_k$ in $C^\infty_{\mathrm{loc}}$ as $j\to\infty$. Then there is $\varphi:\mathbb{N}\to\mathbb{N}$ with $g_{k,\varphi(k)}\to f$ in $C^\infty_{\mathrm{loc}}$ as $k\to\infty$.

Here $u_k\to u$ in $C^\infty_{\mathrm{loc}}$ means: for every $m\in\mathbb{N}$ and every compact $K\subset E$, the $m$-th Fréchet derivatives iteratedFDeriv ℝ m u_k converge to iteratedFDeriv ℝ m u uniformly on $K$. No smoothness of the functions is assumed. This is the step that combines two successive approximation procedures (first $f_k$, then $g_{k,j}$) into one approximating sequence of $f$.