Is $C_{c}^{\infty}[a,b]$ a complete metric space?

Question

Is the space of smooth functions with compact support in an interval $[a,b] \subset \mathbb{R}$, with the metric $\rho(\varphi_{1},\varphi_{2})= \sum_{n=0}^{\infty}2^{-n}\frac{\left \| \varphi_{1} - \varphi_{2} \right \|_{n}}{1 + \left \| \varphi_{1} - \varphi_{2} \right \|_{n}}$ where $\left \| \varphi\right \|_{n} = \sum_{j=0}^{n}\sup_{x \in [a,b]}|\varphi^{(j)}(x)|$ complete?

Answer 1

Every $C^{\infty}$ function on $[a,b]$ has compact support because $[a,b]$ is compact. This space is complete and the proof uses the following result repeatedly: if $f_n \to f $uniformly and $f_n' \to g$ uniformly then $f$ is differentiable and $g=f'$.