I'm reading Remark 12.14.c from textbook Analysis I by Amann/Escher.
$\mathbb{K}_{k}[X]$ is the ring of polynomials whose degrees are less than or equal to $k$.
Here is Formula 12.15:
where the divided difference operator $\triangle_{h}$ of length $h$ is defined by
Here is Remark 8.19 (c):
I follow authors' hint but still no close to the proof that the function $\mathbb{N} \rightarrow \mathbb{K}, \quad n \mapsto p\left(x_{0}+h n\right)$ is an arithmetic sequence of order $k$.
My attempt:
Lemma: Let $E$ be a vector space over field $\mathbb{K}$. On $E^{\mathrm{N}}$ define the difference operator $\triangle$ by $\triangle f_{n} :=f_{n+1}-f_{n}$ for all $n \in \mathbb{N}$ and $f \in E^{\mathbb{N}}$. Then we have $$\triangle^{k} f_{n} = \sum_{j=0}^{k}(-1)^{k-j} {k \choose j} f_{n+j}, \quad k, n \in \mathbb{N}, \quad f \in E^{\mathbb{N}}$$
By 12.15 and Remark 8.19 (c), $$p =\sum_{i=0}^{k} \frac{\triangle_{h}^{i} p\left(x_{0}\right)}{i !} \prod_{j=0}^{i-1}\left(X-x_{j}\right)$$ and thus $$p(x_{n+i}) =\sum_{i=0}^{k} {n+i \choose i} \triangle^i p(x_0)$$
By lemma, $$\triangle^{k} f_{n} = \sum_{i=0}^{k}(-1)^{k-i} {k \choose i} p(x_{n+i})$$
From here, I'm stuck at proving $\triangle^{k} f_{n}$ is a constant.
Could you please get me some hints to finish the proof?