Let $X$ be a function space completed with converge topology (if possible $X$ is normed) such that $C_0^\infty(\mathbb{R}^n)\hookrightarrow X\hookrightarrow L^1(\mathbb{R}^n)$ is continuous dense embedding.
We can contruct a sequence in $C_0^\infty(\mathbb{R}^n)$ by mollified $f\in X$, for example, let $\rho_\delta (x)=\exp(\frac{\delta^2}{|x|^2-\delta^2})1_{B(0,\delta)}$, therefore $(f1_{B(0,R)})*\rho_\delta\in C_0^\infty(\mathbb{R}^n)$.
In general situation, for all $f\in X$, when $\delta_n\to0^+, R_n\to+\infty$, we have $(f1_{B(0,R_n)})*\rho_{\delta_n}\xrightarrow{X}f$.
But is that always true, can we find a space $X$ such that $C_0^\infty(\mathbb{R}^n)$ is dense in $X$, but there is an element $f\in X$, its mollified function never converge to f.