Let $M = \{1, \dots, n\}$ be a metric space with the metric $d$ and, in $\Omega = M^{\mathbb{N}}$, define $\tilde{d}(x, y) = \sum_{k=1}^{+\infty} \frac{d(x_k, y_k)}{2^k}$.
We say that $f\colon \Omega \rightarrow \mathbb{R}$ depens only on finite coordinates if there exist $m \in \mathbb{N}$ such that $f(x_1, x_2, \dots) = f(x_1, \dots, x_m)$.
I'm trying to show that f $f: \Omega \rightarrow \mathbb{R}$ depends only on finite coordinates, then $f$ is $\alpha$-Holder