Convergence to zero in C^∞ on a cylindrical end for pairs

Let $F,G$ be real normed spaces and $f:\mathbb{R}^2\to F$, $g:\mathbb{R}^2\to G$. If $f\to0$ and $g\to0$ in $C^\infty$ as $s\to+\infty$, uniformly in $t$, then so does the pair $(f,g):\mathbb{R}^2\to F\times G$.

It lets several quantities on a cylindrical end that converge in $C^\infty$ be bundled into one vector-valued function.

Formalization Note TendstoZeroCInfty f means: $f$ is $C^\infty$ on some half-plane $(s_0,\infty)\times\mathbb{R}$, and for every $m$ the $m$-th Fréchet derivative $D^mf(s,t)$ tends to $0$ uniformly in $t$ as $s\to+\infty$.