I was attempting to prove the Taylor Theorem using induction, with the following inductive proposition: Let $P(n)$ the proposition that if $f\colon [a,b]\to\mathbb{R}$ be such that $f,f',f'',\cdots, f^{(n)}$ are continuous on $[a,b]$ and $f^{(n+1)}$ exists on $(a,b)$, then for any $x_0 \in [a,b]$, for any $x \in [a,b]$, there is some $c$ between $x_0$ and $x$ such that $$f(x) = \sum_{i=0}^{n-1} (f^{(i)}(x_0) \frac{(x-x_0)^{i}}{i!}) +\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}(c)$$
The base case is to that $f(x) = f(x_0) + f'(c)(x-x_0)$ for some $c \in (x_0, x)$. To prove this I used the mean value theorem. Since $f$ is continuous on $[x_0,x]$ and differentiable on $(x_0,x)$, for some $c \in (a,b)$, $f(x)-f(x_0) = f'(c)(x-x_0)$, so $f(x) = f(x_0) + f'(c)(x-x_0)$, and thus $P(1)$ holds.
Now suppose for some $k \in \mathbb{N}$ that $P(k)$ is true. Thus $$f(x) = \sum_{i=0}^{k-1} (f^{(i)}(x_0) \frac{(x-x_0)^{i}}{i!}) +\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}(c)$$. We will use the mean value theorem again on $f^{(n+1)}(c)$ to find $f^{(n+1)}(c) = f^{(n+1)}(x_0) + f^{(n+2)}(c')(x-x_0)$. It follows that $f^{(n+1)}(c)\frac{(x-x_0)^{n+1}}{(n+1)!}=f^{(n+1)}(x_0)\frac{(x-x_0)^{n+1}}{(n+1)!}+f^{(n+2)}(c')\frac{(x-x_0)^{n+2}}{(n+1)!}$, so substituting this into $\sum_{i=0}^{k-1} (f^{(i)}(x_0) \frac{(x-x_0)^{i}}{i!}) +\frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}(c)$ gives $$f(x) = \sum_{i=0}^{k} (f^{(i)}(x_0) \frac{(x-x_0)^{i}}{i!}) +f^{(n+2)}(c')\frac{(x-x_0)^{n+2}}{(n+1)!}$$
Obviously this is a little off. It seems like I have actually proven that $$f(x) = \sum_{i=0}^{n-1} f^{(i)}(x_0) (x-x_0)^{i} +(x-x_0)^{n+1}f^{(n+1)}(c)$$
which I think is not true. Where did I go wrong in my proof, and what can I do to fix it?