Convergence of fractional laplacian in $\mathcal{S}_s$

Question

Let $s\in(0,1)$, let: $$ \mathcal{S}_s=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha f(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}.$$ The linear space $\mathcal{S}_s$ is endowed with the countable family of seminorms: $$ [f]_\alpha:=\sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha f(x)|,\quad\forall\alpha\in\mathbb{N}_0^n. $$ Consider the Schwarz space $\mathcal{S}(\mathbb{R}^n)$ endowed with the countable family of seminars: $$ [f]_{N,\beta}=\sup_{x\in\mathbb{R}^n}(1+|x|^N)|\partial^\beta f(x)|,\quad\forall\beta\in\mathbb{N}_0^n,N\in\mathbb{N}. $$ I know that: $(-\Delta)^s f\in \mathcal{S}_s,\forall f\in\mathcal{S}$, and: $\mathcal{S}(\mathbb{R}^n)\subset\mathcal{S}_s$. Is true that: $f_j\to f$ in $\mathcal{S}(\mathbb{R}^n)$ implies $(-\Delta)^sf_j\to (-\Delta)^sf$ in $\mathcal{S}_s$? I have no idea. Any help would be appreciated.