 Density of test functions in distributions

Every distribution is a limit of test functions in weak convergence of distributions. With a mollifier ${\rho }_{\epsilon }$ and an expanding smooth cutoff function $\chi (x/j)$, the functions $\chi (x/j)(u\ast {\rho }_{1/j})(x)$ have compact support and converge to $u$. For each fixed test, the cutoff eventually equals one on its support, while the reflected mollifier approximates that test with all derivatives in a common compact set.