Real function/Taylor formula/(n+1)-times continuously differentiable/Estimate for error/Fact

Suppose that ${}I$ is a bounded closed interval,

$f\colon I\longrightarrow \mathbb {R}$

is an ${}(n+1)$-times continuously differentiable function, ${}a\in I$ an inner point, and ${}B:={\max {\left(\vert {f^{(n+1)}(c)}\vert ,c\in I\right)}}$.

Then, between ${}f(x)$ and the ${}n$-th

Taylor polynomial, we have the estimate

${}\vert {f(x)-\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}}\vert \leq {\frac {B}{(n+1)!}}\vert {x-a}\vert ^{n+1}\,.$

Proof, Write another proof