I am trying to do the following question
Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$.
In general in $\mathbb{Q}_p$ what is the stronger condition, to be p-adically convergent or p-adically Cauchy?
I am trying to do the following question
Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$.
In general in $\mathbb{Q}_p$ what is the stronger condition, to be p-adically convergent or p-adically Cauchy?
I guess the thing to notice is that in the $p$-adics we have a geometric series \[ \frac{1}{1 - p} = 1 + p + p^2 + \cdots; \] you can prove this as before, noting that |p| < 1 in our new absolute value. With $p = 7$ the left side is $-1/6$, so the partial sums of the right side form a sequence in $\mathbf Z$ which converges to that number.