Let $\mathcal{C}_{0}^{\infty}(\Omega)$ be the space of smooth functions with compact support in open $\Omega \subset \mathbb{R}^{n}$. Define the Hermitian inner product $(f|g)_{k} := \sum_{|\alpha| \leq k} \int_{\Omega} \partial^{\alpha} f(x) \bar{\partial^{\alpha} g(x)} dx$, $f,g \in \mathcal{C}_{0}^{k}(\Omega)$.
The goal is to prove that $\mathcal{C}_{0}^{\infty}(\Omega)$ is not complete with the Hermitian inner product defined above. I want to find a Cauchy sequence to show that it doesn't converge in that space, but do not know how to design one. Can anyone give some instructions? Thanks!