Theorem 25.5.1 (Taylor’s Formula, Lagrange Remainder).label Let $-\infty < a < b < \infty$, $E$ be a separated locally convex space, $S \subset [a, b]$ be at most countable, $n \in \natp$, and $f \in C^{n}([a, b]; E)$ be $(n+1)$-fold differentiable on $[a, b] \setminus N$, then
\[f(b) - f(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k}f(a)(b - a)^{k}\]
is contained in the closed convex hull of
\[\bracs{D^{n+1}f(s)(t - a)^{n+1} | s \in (a, b) \setminus N, t \in [a, b]}\]
Proof [Theorem 4.7.1, BS17]. If $n = 0$, then the theorem is the Mean Value Theorem.
Suppose inductively that the theorem holds for $n$. Let
\[g: [a, b] \to E \quad t \mapsto f(t) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(t - a)^{k}\]
then for any $t \in (0, 1)$,
\begin{align*}Dg(t)&= Df(t) - \sum_{k = 1}^{n+1}\frac{1}{(k-1)!}D^{k}f(a)(t - a)^{k-1}\\&= Df(t) - Df(a) - \sum_{k = 1}^{n}\frac{1}{k!}D^{k+1}f(a)(t - a)^{k}\end{align*}
by the Mean Value Theorem,
\[g(b) - g(a) = f(b) - f(a) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(t - a)^{k}\]
is contained in the closed convex hull of
\[\bracs{\braks{Df(t) - Df(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k+1}f(a)(t - a)^{k}}(b - a) \bigg | t \in [a, b]}\]
By the inductive hypothesis applied to $Df$, for any $t \in [a, b]$,
\[Df(t) - Df(a) - \sum_{k = 1}^{n} \frac{1}{k!}D^{k+1}f(a)(t - a)^{k}\]
is contained in the closed convex hull of
\[\bracs{D^{n+2}f(s)(t - a)^{n+1} | s \in (a, t) \setminus N}\]
Therefore
\[f(b) - f(a) - \sum_{k = 1}^{n+1}\frac{1}{k!}D^{k}f(a)(b - a)^{k}\]
is contained in the convex hull of
\[\bracs{D^{n+2}f(s)(t - a)^{n+2} | s \in (a, t) \setminus N, t \in [a, b]}\]
$\square$