Prove that the given space $H^1$ is not complete.

Question

Let $ \Omega \subset \mathbb{R}^n$ be open and bounded. For $u, v \in C^1(\overline{\Omega}) $, define $$ (u,v)_{H^1} = \int_{\Omega} u(x)v(x)dx + \int_{\Omega}\nabla u(x)\cdot \nabla v(x)dx. $$ I want to show that $(C^1(\Omega), (\cdot,\cdot)_{H^1})$ is an inner product space but not complete. I've already proved that $(u,v)$ is an inner product function, but I have no clue on how to prove that it is not complete.