Given are $n \in \mathbb N, h > 0$ and $a \in C^{n+1}(0, h)$. Furthermore, let $(x_k)_{k = 0, \dots, n}$ be a monotonic decreasing sequence of positive numbers with: $$x_0 \leq h \;\;\; \text{ and } \;\;\; \frac{x_{k + 1}}{x_k} \leq \rho < 1 \;\;\; \forall k \geq 0$$ Then it holds for the interpolation polynomial $p \in \mathbb P_N$ through the points $(x_k, a(x_k))_{k = 0, \dots, n}$ that: $$\lvert a(0) - p(0) \rvert \leq \frac{\lvert \lvert a^{(n+1)}\rvert \rvert_{\infty, [0, h]}}{(n + 1)!}h^{n+1}$$ (Note that we define $a(0) = \lim_{x \searrow 0} a(x)$.)
The first thing to show in would be that there exist $b_1, \dots, b_n \in \mathbb R$ and $\xi_x \in (0, h)$, such that: $$p(0) = a(0)\left( \sum_{i = 0}^n l_i^{(n)}(0) \right) + \sum_{j = 1}^n b_j \left( \sum_{i = 0}^n x_i^j l_i^{(n)}(0) \right) + \frac{a^{(n+1)}(\xi_x)}{(n+1)!} \left( \sum_{i = 0}^n x_i^{n+1} l_i^{(n)}(0) \right)$$
However, I am already stuck here. I'm pretty sure that I should make use of the Taylor polynomial, which gives me that there exist an $\xi_x \in (0, h)$, such that: $$p(0) = \sum_{i = 0}^n \frac{p^{(i)}(h)}{i!}(-h)^i + \frac{p^{(n+1)}(\xi_x)}{(n+1)!}(-h)^{n+1}$$
But I don't know how to really proceed from here on out. I would appreciate help or even a reference to a work where this proof is discussed!
The interpolation error is
$$p(x) - a(x) = \frac{a^{(n+1)}(\zeta)}{(n+1)!}\prod_{i=0}^n (x - x_i)$$
where $\zeta\in [0, h]$, you just put $x= 0$ and use the fact $|x_i|\le \rho^i h$, then $\prod_{i=0}^n |x_i| \le h^{n+1} \rho^{n(n+1)/2} < h^{n+1}$.