Convergence in distribution to delta

Question

Let us assume $f_n\in S'(\mathbb{R}^k)$ (Schwartz space) such that $f_n\to\delta_t$ in distribution sense. That is, $$\int_{\mathbb{R}^k}f_n(s)F(s)\mathrm{d}s\to F(t)$$ for all $F\in S(\mathbb{R}^k)$. Fix $g\in C^\infty(\mathbb{R}^k)\cap L^2(\mathbb{R}^k)$ (with no necesarily compact support). Is it true that $\int_{\mathbb{R}^k}f_n(s)g(s)\mathrm{d}s\to g(t)$?