Borel reducibility and finitely Holder(\alpha) embeddability

Let $(X_n,d_n),n\in \mathbb{N}$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in {\prod }_{n\in \mathbb{N}}{X}_n$, let $(x,y)\in E((X_n){n\in\Bbb N};p)\Leftrightarrow\sum{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n{)}_{n\in \mathbb{N}};p)$, we show it is closely related to finitely H"older($\alpha$) embeddability between pseudo-metric spaces.