Taylor's Theorem/One Variable/Proof by Rolle's Theorem

Proof
Let the function $g$ be defined as:
 * $g \left({t}\right) = R_n \left({t}\right) - \dfrac {\left({t - a}\right)^{n + 1} } {\left({x - a}\right)^{n + 1} } R_n \left({x}\right)$

Then:
 * $g^{\left({k}\right)} \left({a}\right) = 0$

for $k = 0, \dotsc, n$, and $g \left({x}\right) = 0$.

Apply Rolle's Theorem successively to $g, g', \dotsc, g^{\left({n}\right)}$.

Then there exist:
 * $\xi_1, \ldots, \xi_{n + 1}$

between $a$ and $x$ such that:
 * $g' \left({\xi_1}\right) = 0, g'' \left({\xi_2}\right) = 0, \ldots, g^{\left({n + 1}\right)} \left({\xi_{n + 1} }\right) = 0$

Let $\xi = \xi_{n + 1}$.

Then:
 * $0 = g^{\left({n + 1}\right)} \left({\xi}\right) = f^{\left({n + 1}\right)} \left({\xi}\right) - \dfrac {\left({n + 1}\right)!} {\left({x - a}\right)^{n + 1} } R_n \left({x}\right)$

and the formula for $R_n \left({x}\right)$ follows.