Let $M_X = (X, d_X)$ and $M_Y = (Y, d_Y)$ each be a D1107: Metric space.
Let $f : X \to Y$ be a D47: Lipschitz map with respect to $M_X$ and $M_Y$.

Since $f$ is Lipschitz, there is $M \geq 0$ such that $d_Y(f(x), f(y)) \leq M d_X(x, y)$ for every $x, y \in X$. This is precisely the condition for Hölder-continuity with $\alpha = 1$. $\square$