Let $f \in L^1_{loc}$, and define $u_f \in \mathcal{D}'(\mathbb{R}^n)$ to be any distribution of the form $$\langle u_f, \phi\rangle = \int f\phi.$$
I would like to show that the collection of all such $u_f$ is dense in the space of distributions $\mathcal{D}'(\mathbb{R}^n)$.
The motivation for this question came form this Wiki on distributions, which provides a proof of the formula for the differential operator acting on distributions. What bothers me is that the proof relies on representing the distribution as an integral, but clearly not every distribution can be represented this way (take the Dirac-$\Delta$ function for example). The article hints that collection of $u_f$ mentioned above is dense in the space of distributions, so it suffices to consider this case and from there I assume one can make a limiting argument.
How would one go about proving this?