Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 12E / Solution

For $q\in \mathbb{Q}\setminus \{0\}$ write $q=p^{k}u/v$, where $k\in \mathbb{Z}$ and neither integer $u$ nor integer $v$ is divisible by $p$. Define the P-adic valuation $v_p(q)=k$ and the p-adic absolute value $|q{|}_p=p^{-k}$, with $|0{|}_p=0$. The p-adic metric isFactoring out the smaller power of $p$ from two rational numbers shows $v_p(q+r)\ge \min \{v_p(q),v_p(r)\}$, with the usual convention $v_p(0)=+\infty$. Consequently the ultrametric inequality givesTogether with symmetry, positivity and separation of points, this verifies that $d_p$ is a metric.

For the given geometric partial sums, use the finite geometric series identity. Since $p$ does not divide $p-1$,Thus the limit is the rational number $\boxed{1/(1-p)}$, even though the partial sums diverge in the ordinary real metric.

If $|a{|}_p>|b{|}_p$, the ultrametric inequality gives $|a+b{|}_p\le |a{|}_p$. Applying it to $a=(a+b)-b$ gives $|a{|}_p\le \max \{|a+b{|}_p,|b{|}_p\}$. Since $|b{|}_p<|a{|}_p$, this forces $|a+b{|}_p=|a{|}_p$. Interchanging $a,b$ handles the other unequal case, provingFinally, let $b$ be outside the open ball $B(a,\delta )$, so $d_p(a,b)\ge \delta$. If $d_p(b,c)<\delta$, the preceding unequal-norm identity gives $d_p(a,c)=d_p(a,b)\ge \delta$. Therefore $B(b,\delta )$ also lies outside $B(a,\delta )$. Its complement is open, so every p-adic open ball is also closed, that is, a clopen set.