Besov spaces: Regularization by convolution

Consider either a Besov space $$B_{p,q}^s$$ or a Triebel-Lizorkin space $$F_{p,q}^s$$ with parameters $$p,q \in [1,\infty)$$ and $$s \in \mathbb{R}$$. It is known that the Schwartz space $$\mathcal{S}(\mathbb{R})$$ is dense in $$A_{p,q}^s$$ for $$A \in \{B,F\}$$. I was looking for a "concrete" approximation. From my perspective, the most natural idea is to use convolutions: Fix $$f \in A_{p,q}^s$$ with $$A \in \{B,F\}$$. For $$\chi \in C_c^{\infty}(\mathbb{R}^n)$$ with $$\chi \geq 0$$ and $$\int \chi(x) \, dx =1$$, define $$\chi_k(x) := k^d \chi(kx)$$. Then

$$f_k := f* \chi_k$$

should converge to $$f$$ in $$A_{p,q}^s$$, right? Moreover, $$f_k$$ is a Schwartz function, and we are done. Does anybody know a reference for this result? I checked the books by Triebel and also the book by Sawano, but I couldn't find it (probably because it is "obvious").