Given the integers and a prime $p$. I thought I had successfully shown that $\mathbb{Z}$ was compact with respect to the metric $|\cdot |_p$, by showing that the open ball centered at zero contained all integers with more than a certain number of factors of $p$, and then showing that the remaining integers took on a finite number of possible p-adic absolute values and thus fell into a finite number of balls.
Now if the integers are compact with respect to $|\cdot |_p$, then that means they are complete with respect to $|\cdot |_p$.
But then I read that the p-adic integers $\mathbb{Z}_p$ are defined to be the completion of the integers with respect to $|\cdot |_p$, and include in their completion all the rational numbers with p-adic absolute value less than or equal to one. So this means that the integers with respect to the p-adic metric are not complete, and thus not compact, and hence there must be something wrong with my proof, correct?
Edit: Ok upon typing this up I realized that my proof is most likely wrong as there's no reason to conclude that two elements with the same absolute value are necessarily in the same ball.
Best Answer
Jyrki showed you a specific example of a cauchy sequence not converging in $\mathbb{Z}$, but there is a wholly more dramatic answer. Let's suppose that $\mathbb{Z}$ was complete. Then, every infinite series $\sum_{n=0}^{\infty}a_np^n$ with $a_n\in\{0,1,\ldots,p-1\}$ would converge.(because each such series has partial sums that are Cauchy). Moreover, two such infinite series are equal if and only if their coefficients (of $p^n$) are equal (just check their valuations). Thus, we'd have an injection $(\mathbb{Z}/p\mathbb{Z})^\mathbb{N}\to \mathbb{Z}$ which is problematic due to carinality issues.
This also shows that $\mathbb{Q}$ is not complete