Taylor Theorem Differentiable Function I've got the following question

Let $g:\mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable
  function satisfying $g(0)=1, \  g'(0) = 0$ and $g''(x) - g(x) = 0$,
  for all $x \in \mathbb{R}$
i) Prove that $g$ has derivatives of all orders.
ii) Fix $x \in \mathbb{R}$. Show that there exists $M > 0$ such that for all $n \in \mathbb{N}$ and all $\theta \in (0,1)$
$|g^{(n)}(\theta x)| \leq M$
iii) Find the coefficients of the Taylor expansion of $g$ about $0$, and prove that this expansion converges to $g(x)$ for all $x \in
 \mathbb{R}$

I've done ( i ) and noted that $g^{(k)}(0) = 1$ if $k$ is even and $g^{(k)}(0) = 0$ if $k$ is odd. I'm struggling with the last two parts. For ( ii ) I've shown that, via Taylor's Theorem, we get
$g(x) = \displaystyle\sum_{k=0}^{n-1} \dfrac{g^{(k)}(0)}{k!} x^k + \dfrac{g^{(n)}(\theta x)}{n!} x^n$
But I don't see how to get an upper bound on $\dfrac{n! g(x)}{x^n} - \dfrac{n!}{x^n} \displaystyle\sum_{k=0}^{n-1} \dfrac{g^{(k)}(0)}{k!} x^k$ for some fixed $x \in \mathbb{R}$
I've also realised that $\displaystyle\lim_{n \to +\infty}\displaystyle\sum_{k=0}^{n-1} \dfrac{g^{(k)}(0)}{k!} x^k = \cosh(x)$ 
Any help greatly appreciated. 
Thanks!
 A: ii) Let $x \in \mathbb{R}$. Since $g$ is twice differentiable, it is $C^1$. Let $M = \sup_{\lambda \in [0,1]} \max(|g(\lambda x)|,|g'(\lambda x)|)$.
Since $g^{(2)}(x) = g(x)$, we have $g^{(2n)}(x) = g(x)$ for all $n$. Similarly, since $g^{(3)}(x) = g^{(1)}(x)$, we have $g^{(2n+1)}(x) = g^{(1)}(x)$ for all $n$. Consequently it follows that $|g^{(n)}(\theta x)| \le M$ for all $n$ and for all $\theta \in (0,1)$.
iii) The Taylor coefficients about zero follow from the above formula for $g^{(n)}$. In particular, $g^{(2n)}(0) = 1$, $g^{(2n+1)}(x) = 0$.
Following your notation above, to show that the Taylor series converges to $g(x)$, you need to show that for all $\epsilon>0$, we can find $N$ such that for $n \ge N$, $$\left|g(x)-\sum_{k=0}^{n-1} \dfrac{g^{(k)}(0)}{k!} x^k\right| < \epsilon.$$ From your estimate above, we have $$\left|g(x)-\sum_{k=0}^{n-1} \dfrac{g^{(k)}(0)}{k!} x^k\right| = \left|\dfrac{g^{(n)}(\theta x)}{n!} x^n\right| \le M \frac{|x|^n}{n!},$$ and since $\lim_n \frac{|x|^n}{n!} = 0$, it is clear that we can find such a $N$. Hence the Taylor series converges.
And, as you have noted, we have $g(x) = \cosh x$.
