$p$(ain)-adic number sequence I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH!
Jokes aside, I am working on this question

Show that the sequence $(3,34,334,3334,.....)$ is equal to $2/3$ in $\hat{\mathbb{Z}}_5$

I assume they mean it converges to $2/3$ in the $5$-adic integers by the question. My big issue is that I am struggling to properly attack it. One question immediately comes to mind is: does $334$ mean

$334=3\cdot 5^0+3\cdot 5^1+4\cdot 5^2$

as all $p$-adic numbers can be written as such, or what does it exactly mean? If so, how would I go on exactly demonstrating this? Their initial peculiarity, especially with increased number decreases distance is throwing me off quite a bit.

Here, the notation is in the usual base $10$, so the number $a_n = 33\ldots34$ simply denotes
$$a_n = 3 \times 10^n + 3 \times 10^{n-1} + \ldots + 3 \times 10 + 4 = 1 + 3\sum_{k = 0}^n 10^k = 1 + \frac{10^{n+1}-1}{3}$$
Now, in the $5$-adic topology, the sequence $(10^n)_{n \ge 0}$ goes to $0$ at infinity because $\left|10^n\right|_5 = \frac{1}{5^n} \longrightarrow 0$, so you get
$$\lim_{n \to \infty} a_n = 1 + \frac{0-1}{3} = \frac{2}{3}$$
