Is the set of all infinite sequences of natural numbers which are eventually constant countable? I know that the set of all infinite sequences with finite length is countable, this set seems like its just countable copies of the set of all infinite sequences with finite length. How to show this rigorously?

If you're already at the point where you accept that the finite sequences are countable, then you can simply say that the eventually constant sequences are accounted for by taking any finite sequence and then concatenating it with an infinitely repeated value.
I.e, every eventually constant sequence is equal to some $s^\frown x$ where $s\in\mathbb{N}^{<\mathbb{N}}$ and $x=\langle n,n,n,\ldots\rangle$ for some $n\in\mathbb{N}$.
Thus, you've got an injection from the eventually constant sequences into $\mathbb{N}^{<\mathbb{N}}\times \mathbb{N}$, which is countable since $\mathbb{N}^{<\mathbb{N}}$ and $\mathbb{N}$ are countable.
