Let $\{f_k\}_{n \in \mathbb N} \subset S(\mathbb R^n)$ converging to $f \in S(\mathbb R^n)$ in the $S(\mathbb R^n)$ topology.
Is it true that $\int_{\mathbb R^n} f_k$ converges to $\int_{\mathbb R^n} f$?
My attempt
We already know that $S(\mathbb R^n) \subseteq L^1(\mathbb R^n)$ and it's not hard to check that $f_k$ converges pointwise to $f$ (actually uniformly). What's left is to show that $\{f_k\}$ can be dominated by a $L^1(\mathbb R^n)$ function. Any idea?
First, you're using "$n$" for two different things; let's say the $f_n$ are functions on $\Bbb R^d$ instead of $\Bbb R^n$.
I don't see how to use DCT here. Luckily it's easier than that. Not only does $f_n\to f$ uniformly, in fact $(1+|x|)^{d+1}f_n(x)\to(1+|x|)^{d+1}f(x)$ uniformly. Show that $$||f_n-f||_1\le c\sup_{x\in\Bbb R^d}(1+|x|)^{d+1}|f_n(x)-f(x)|$$and you're done.