Taylor expansion in Hoeffding's Lemma proof Hoeffding's Lemma proof uses Taylor expansion with this statement:
From Taylor's theorem, for some $ 0\leq \theta \leq 1$
$  L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $
Why does it use $0$ in the first two terms and $h\theta$ in the last? But as I know they must be same in the Taylor.

This is the mean-value form of Taylor's theorem:
$$f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(c)$$
where $c$ is between $0$ and $x$
Take $x=h$ and $c=h\theta$
