Let $X$ be a Polish space. Let $B$ be a Borel subset of $X \times X$ with the following property: $$\forall x \in X \ \left|\left\{y : (x,y) \in B\right\}\right| = \aleph_0.$$ Show that there exists a sequence of Borel sets $B_0, B_1, \dots, B_n, \ldots \subset X \times X$ such that $B = \bigcup_{n \in \mathbb N} B_n$ and $$\forall n \in \mathbb N \ \forall x \in X \ \left|\left\{y : (x,y) \in B_n\right\}\right|=1.$$
I tried to enumerate elements of each set $\left\{y : (x,y) \in B\right\} = \{x_n : n \in \mathbb N\}$ and define $B_n = \{(x,x_n) : x \in X\}$. I am not sure whether $B_n$s are Borel (I would guess they're not). I am stuck.