How to prove that $C^N_c (K) = \{{\phi\in C^N(R^N):\operatorname{supp}\phi\subset K}\}$ is a Banach space?

Question

I am trying to prove the next claim-

Let K be a compact set, and N a nonnegative integer. then $$C^N_c (K) = \{{\phi\in C^N(R^N):\operatorname{supp}\phi\subset K}\}$$ is a Banach space when it is equipped with the norm $$\phi \rightarrow \Sigma_{|\alpha|\le N} \sup|\partial^\alpha \phi |$$ Is there any connection to uniform convergence?