Borel reducibility and finitely Hِlder() embeddability

Let ( X n , d n ) , n ∈ N , be a sequence of pseudo-metric spaces, and let p ≥ 1 . For x , y ∈ ∏ n ∈ N X n , let ( x , y ) ∈ E ( ( X n ) n ∈ N ; p ) ⇔ ∑ n ∈ N d n ( x ( n ) , y ( n ) ) p < + ∞ . For Borel reducibility between equivalence relations E ( ( X n ) n ∈ N ; p ) , we show it is closely related to finitely Hölder( α ) embeddability between pseudo-metric spaces.