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. 

Suggested by @Luke's comment, we can argue like this. First, as you likely know, a set $V$ is countable if and only if it can be written as as a sequence $\{s_0, s_1, \dotsc \}$. Suppose that your set $V$ was countable. Then we could describe it as a sequence of infinite strings
$$\begin{align}
s_0 &= a_{00}\,a_{01}\,a_{02}\dotsm \\
s_1 &= a_{10}\,a_{11}\,a_{12}\dotsm \\
s_2 &= a_{20}\,a_{21}\,a_{22}\dotsm 
\end{align}$$
and so on, where each $a_{ij} \in \{0,1,2\}$. Since $V$ was assumed to be countable, then by definition, this enumeration must contain all strings in $V$. Now define a string $\hat{s}=d_0d_1d_2\dotsc$ by
$$
d_i = \begin{cases}
        0 &\text{if $a_{ii} = 1$}\\
        1 &\text{if $a_{ii} = 2$}\\
        2 &\text{if $a_{ii} = 0$}
       \end{cases}
$$
Now we see that $\hat{s}$ isn't in the enumeration $s_i$: It can't be $s_0$ since it differs from $s_0$ in the zero-th digit, $a_{00}$; it can't be $s_1$ since it differs from the first digit, $a_{11}$, and so on, so it's a string that is none of the strings $s_0, s_1, \dotsc$. However, our assumption was that there was an enumeration that consisted of all strings over $\{0,1,2\}$, meaning that $V$ couldn't possibly be countable.
This, by the way, is known as Cantor's diagonal argument. I wish I could claim it as my invention, but Cantor came up with it before almost anyone on SE was born.

Your construction was a correct correspondence between $\mathbb{N}$ and all finite strings over $\{0,1,2\}$. Given any $n\in\mathbb{N}$, your correspondence will give a string of $k$ digits, where $k$ is about $\log_3 n$, but notice that we're not dealing with finite strings when we're talking about $V$. If it's any comfort, this is a subtle point that trips a lot of people up (and has led to enormously long and frustrating threads in internet discussion fora).    
