The functions $S$ and $C$ are defined as follows: both are infinitely differentiable, $\frac{d}{dx}S(x) = C(x), \frac{d}{dx}C(x) = S(x)$, and $S(0) = 0, C(0) = 1$.
So, clearly the Taylor expansion of this function on the interval $[a, x]$ with $a = 0, x \in \mathbb{R}$ and $c \in [|-x|, |x|]$ is $$S(x) = \sum_{n=0}^{k-1}\frac{x^n}{n!}S^{(0)}(0) + \frac{x^k}{k!}S^{(k)}(c)$$
My question is how do you show that $\lim_{k\to\infty} \frac{x^k}{k!}S^{(k)}(c_k) = 0$, where the term in the limit is the remainder term in the Taylor expansion of $S(x)$, and each $c_k$ is between $0$ and $x \in \mathbb{R}$?
This is what I've done so far: I have determined that $\lim_{k\to\infty}\frac{x^k}{k!} = 0$, so all that remains is to prove that $\lim_{k\to\infty} S^{(k)}(c_k)$ exists. This is where I'm having trouble.
I've attempted to use Cauchy's Mean Value Theorem and the normal formulation of Mean Value Theorem to come up with expressions for $S^{(k)}(c_k)$, but the oscillatory nature of $S$ means I always have two different expressions for each parity of $k$. I've also tried setting maximum bounds $M_k$ for each $S^{(k)}(c_k)$, but that doesn't help either, since I have no way of knowing whether the sequence of $M_k$'s is bounded or not. What approach am I not seeing?
Any help is much appreciated!