Thread: Prove Or Disprove Completeness of Z

1. Prove Or Disprove Completeness of Z

For this problem, we're dealing with the set of integers with the metric $d(x,y)=\frac{1}{5^k}$ where $k$ is the highest power of of 5 that divides $x-y$.

I'm trying to either prove or disprove that this is a complete space. I know I have to either come up with a Cauchy sequence of integers that does not converge in the integers with respect to $d$, or show that every Cauchy sequence in the integers converges in the integers with respect to $d$. I'm just not seeing which one is the case. Does anyone have a hint to get me going in the right direction?

2. Originally Posted by mathematicalbagpiper
For this problem, we're dealing with the set of integers with the metric $d(x,y)=\frac{1}{5^k}$ where $k$ is the highest power of of 5 that divides $x-y$.

I'm trying to either prove or disprove that this is a complete space. I know I have to either come up with a Cauchy sequence of integers that does not converge in the integers with respect to $d$, or show that every Cauchy sequence in the integers converges in the integers with respect to $d$. I'm just not seeing which one is the case. Does anyone have a hint to get me going in the right direction?
Think about the sequence $(x_n)$, where $x_n = 5^n$. Then read about p-adic numbers (with p=5 here).