Consider the set $C^k(\mathbb{R^n},\mathbb{R^m})$ in the metric $\rho$ where $\rho$ is defined by $$\rho(f,g)=\sup\{d(f(x),g(x)),d(f^1(x),g^1(x)),\dots ,d(f^k(x),g^k(x))|x\in \mathbb{R^n}\}.$$
How can I show, $(C^k(\mathbb{R^n},\mathbb{R^m}),\rho)$ is not a complete metric space? Also if we consider $C^\infty(\mathbb{R^n},\mathbb{R^m})$ in the metric $\rho$,where $$\rho(f,g)=\sup\{d(f(x),g(x)),d(f^1(x),g^1(x)),\dots,d(f^k(x),g^k(x)),\dots|x\in \mathbb{R^n}\}$$ then how to show that $(C^\infty(\mathbb{R^n},\mathbb{R^m}),\rho)$ is not complete? I am completely stuck. Please help!