/sci/ - Science & Math

Maths warmup exercise!
Let [math]X = \{0,1\}^\mathbb{N}[/math]. For [math]x,y \in X[/math], define [math]x\sim y \iff \lim_{n \to \infty} e(x,y,n)/n = 0[/math], where [math]e(x,y,n)= |\{ k \leq n : x_k \neq y_k \}|[/math]. It's easy to verify that [math]\sim[/math] is an equivalence relation.
Question: prove or disprove that [math] |X / \sim| = |X|[/math]