$C^n[a,b]$ is a Banach algebra

Question

Prove that $C^n[a,b]$ with pointwise multiplication of functions and a norm defined as $\displaystyle\|x\| = \sum_0^n \dfrac{\|x^{(k)}\|_{\infty}}{k!}$ is a Banach algebra. As I understand I need to prove that a) $C^n[a,b]$ is complete, b) it's algebra (?) and c) $\|x\cdot y\| < \|x\|\cdot\|y\|$, but I'm not sure how to do it...