Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ that satisfies these conditions?

Fix a bijection $\mathbf{N} \leftrightarrow \mathbf{Q}$. For each $x \in \mathbf{R}$ map it to a set of rational numbers $\{x_1,x_2,\dots\}$ corresponding to a sequence that converges to $x$. Call this map $f : \mathbf{R} \to P(\mathbf{Q})$. Then show that $f$ is injective, $f(x)$ is infinite and if $x \ne y$ then $f(x) \cap f(y)$ is finite.

First, get an injection $g : \mathbb{R} \to P(\mathbb{N} \times \mathbb{N})$ satisfying these conditions by setting $g(x) := \{ (n, \lfloor 10^n x \rfloor) : n \in \mathbb{N} \}$. Now, using the fact that $\mathbb{N} \times \mathbb{N}$ is countable, can you see a way to convert $g$ to a function $f$ satisfying the conditions?