Let $X$ be a complete metric space and $\alpha \in (0,1)$. Suppose that for every $x,y \in X$ there exists $z \in X$ s.t. $ d(x,z) \le \frac{1}{2^\alpha}d(x,y), \qquad d(y,z) \le \frac{1}{2^\alpha}d(x,y). $
Then $X$ is Hölderian path-connected, i.e. for every $x,y \in X$ we can find a $\alpha$-Hölderian path $\gamma \colon [0,1] \to X$ such that $\gamma(0)=x$ and $\gamma(1)=y$.
Have you got any ideas on how to solve this problem? To be honest, I do not know how to start. I'm stumped for the hypothesis of completness... How can we use it?
I thank you in advance for any useful idea.