How to prove $C_{c}^{\infty}[a,b]$ a complete metric space

Question

How do you prove 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 $\left \| \varphi\right \|_{n} = \sum_{j=0}^{n}\sup_{x \in [a,b]}|\varphi^{(j)}(x)|$ is complete? I've tried to consider the pointwise limit and use the fact that $\mathbb{R}$ is complete but as yet have been unsuccessful. Apparently the proof uses the following fact repeatedly: if $f_{n} \to f$ uniformly and $f_{n}' \to g$ uniformly, then f is differentiable and $g= f'$