Homework question on convergence using the p-adic metric

Question

I am self-teaching myself about metric spaces (officially a physics student), and have come across the following question:

I am asked to show that the sequence $215, 2015, 20015, 200015... $converges in the $2$-adic metric on $\mathbb{Z}$.

Now the p-adic metric is that induced by the p-adic norm $||n||=p^{-k}$ where $k$ is the highest power of the prime $p$ that divides $n$ (any motivation as to why this would be a useful metric would also be appreciated...)

Now if I choose an odd integer $x=2m+1<215$, and let the terms of the above sequence be $x_n$, then clearly $x_n-x$ is divisible by $2$ for all $n$, and the 2-adic norm of $||x_n-x||$ tends to $0$ for this sequence. So any such x satisfies the condition for the convergence limit o this series, meaning that the series converges to any odd integer $<215$ in $\mathbb{Z}$? Ad yet I have seen a general proof that the convergence limit of a sequence under any metric is unique!

Note: This question has been edited so some of the initial comments no longer apply.

Answer 1

I don't know much about the p-adic metric but won't this sequence converge to $15$?

Namely, if $a_n$ denotes the number $20\ldots 015$, with $n$ zeros, then $a_n-15= 2\cdot 10^{n+2}= 2^{n+3} 5^{n+2}$, so $\|a_n-15\|=2^{-n-3}$.

Answer 2

As the other answer already gives a proof why $x_n$ converges to $15$, I just address your other question. You write

Now if I choose an odd integer $x=2m+1<215$, and let the terms of the above sequence be $x_n$, then clearly $x_n-x$ is divisible by 2 for all $n$,

This is correct (and, by the way, would be true for any odd integer $x$; I don't see why you think the restriction $<215$ would be necessary).

and the 2-adic norm of $||x_n-x||$ tends to 0 for this sequence.

This is not correct, unless $x$ is 15. The $2$-adic norm of $x_n-x$, for an odd integer $x$, is given by $2^{-r}$, if $2^r$ is the highest power of $2$ which divides $x_n-x$. Now you can try out in some examples (use $x=17, 19, 39$) how in these cases, the $2$-adic norm of $x_n-x$ actually goes to 1/2, 1/4, 1/8 respectively. And then, good practice for further studies of $p$-adics, think about why I chose those examples (hint: for these x, what is their $2$-adic distance to 15?)