Determine whether the given set is countable Given the set $V = $ { $w$ | $w$ is an infinite sequence in the form of $a_0a_1a_2\ldots$ with $a_i \in \{0,1,2\}$ for all $i \ge 0$ }
Determine if V is countable. 
I'm assuming V is countable, and have included what I believe is the proof below, I'm wondering if I am correct in my assumption that it is countable, and if the proof is sufficient. 
I have that V is countable, because for each $i \ge 0$ we have a string $w$ that has length $i+1$ and has $3^{i+1}$ permutations. Meaning that if $i = 0$ we could map $1, 2, 3$ to the possible strings, and if $i = 1$ we can map $4, 5, \ldots, 12$ to the 9 strings with length $2$. 
This same principle holds for all $i \ge 0$. Therefore there is a correspondence between $\mathbb{N}$ and V. 

Let $w \in V$ be a string consisting of only 0s and 1s.  We may interpret $w$ as the characteristic function of a set $S_w \subseteq \mathbb{N}$ of natural numbers in the following way:
$$ x \in S_w \text{ if and only if } w_x = 1 . $$
This immediately given a bijection between $V_{0,1}$, the strings with only 0s and 1s, and the power set of the natural numbers.
It follows that $V_{0,1}$ is uncountable.  Since $V$ has $V_{0,1}$ as a subset, $V$ is uncountable.
