For any fixed $C > 0$, prove that $\frac{C^{k}}{k!} \to 0$ as $k \to \infty$

The hint given is to choose an integer $K>2C$ and examine the ratio in the question for $k >K$

I'm not quite sure how the hint applies here. I tried proving that applying the ratio test to $\frac{C^{k}}{k!}$ shows that it is convergent, and then showing that the limsup as $k \to \infty$ is less than 1 and thus lim$\frac{C^{k}}{k!} = 0$, but I don't think the method works given the hint in the question. 

The second part of the question says "Let $f$ be $C^{\infty}$ on [b,c] such that $|f^{(k)}(x) \leq M$ for all $k \in \mathbb{N}$ and $x \in (b,c)$. Given $a \in (c,d)$, show that $P\_{a,k} \to f$ uniformly on [b,c]" and I get how the first result applies here. I'm just unsure how to actually prove that $\frac{C^{k}}{k!} \to 0$ as $k \to \infty$