Taylor's inequality for vector valued functions

Question

I have to show as a general form of 5.19 theorem of baby Rudin that: $| f(\beta) - \sum_{k=0}^{n-1} \frac{f^k(\alpha)(\beta- \alpha)^k}{k!}| \le \frac{(\beta - \alpha)^n f^n(c)}{n!}$. With the same technique in Ruding's book, I put z = f(b) - f(a) and defined $\phi$(t) = z.f(t), and applied taylor's theorem to $\phi$(t) and concluded that $ z.( f(\beta) - \sum_{k=0}^{n-1} \frac{f^k(\alpha)(\beta- \alpha)^k}{k!}) = z.(\frac{(\beta - \alpha)^n f^n(c)}{n!})$ for some c between $\alpha$ and $\beta$. but I don't know how to proceed.