I'm trying to understand the proof of Proposition 1.13 in Tsybakov's book Introduction to Nonparametric Estimation. I'm stuck on the following detail.
Let $f : \mathbb{R} \mapsto \mathbb{R}$ be $\ell$ times continuously differentiable with bounded derivatives. Then, for any two points $x$ and $x_0$ Tsybakov claims we can write:
$$f(x) - f(x_0) = \frac{(f^{(\ell)}(z) - f^{(\ell)}(x_0)) (x - x_0)^\ell}{\ell!}$$
for some $z = x_0 + \tau (x - x_0)$ where $0 \leq \tau \leq 1$. This almost looks like the Lagrange remainder in Taylor's theorem and it almost looks like a higher-order extension of the mean value theorem, but neither seem quite right. All Tsybakov says is that we need to use "Taylor expansion of f." But I'm confused about why the first $\ell - 1$ terms would disappear?
I don't even get the right result in the special case where $\ell = 1$. Here we would have (using the Lagrange remainder):
$$f(x) = f(x_0) + f^{(1)}(z)(x - x_0) \quad \Rightarrow \quad f(x) - f(x_0) = f^{(1)}(z)(x - x_0)$$
which doesn't quite fit the claimed form because I'm missing a subtraction off of $f^{(1)}(x_0)(x - x_0)$.
I'm pasting a small screenshot below from the book. The final step in the sequence below is the one I don't understand.
