The problem
Let $X$ be a set. We define an equivalence relation ~ on $X^{\mathbb{N}}$:
$(a_n)_n$ ~ $(b_n)_n \iff \exists n_0 \in \mathbb{N}, \forall n \geq n_0 : a_n = b_n$ for $(a_n)_n, (b_n)_n \in X^{\mathbb{N}}$.
Show that $|X^{\mathbb{N}}/$~$| = 2^k $ if $ |X| = 2^k$ with $k\geq |\mathbb{N}|$
My attempt
I tried to construct an injection from the equivalence classes to $X$, by mapping the "last element" of the rows from each equivalence class to $X$ but I'm pretty sure this isn't correct. How should I approach this?
Clearly $|X^{\mathbb N} / \sim | \le |X|^{\mathbb N}$. And we actually have that equality holds: By assigning each $x \in X$ to the equivalence class of the function $c_{x} \colon \mathbb N \to \{x \}$, we obtain that $|X| \le |X^{\mathbb N} / \sim |$. Now prove that, under the given assumptions, $|X| = |X|^{\mathbb N}$ and you're done.