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?

Given such a sequence $a:\mathbb{N}\to \mathbb{N}$ define, 
$$ f(a) = (a_0,a_1,a_2,...,a_k,c) $$
(I live is up to you to figure out what that means above). 
Thus, we have constructed an injection $f:\text{const}\to F$, where $F$ are the finite sequences of the natural numbers and $\text{const}$ are the eventually constant sequences. 
