TITLE: Further reading on the $p$-adic metric and related theory.
QUESTION [6 upvotes]: In his book Introduction to Topology, Bert Mendelson asks to prove that
$$(\Bbb Z,d_p)$$
is a metric space, where $p$ is a fixed prime and 
$$d_p(m,n)=\begin{cases} 0 \;,\text{ if }m=n \cr {p^{-t}}\;,\text{ if } m\neq n\end{cases}$$
where $t$ is the multiplicty with which $p$ divides $m-n$. Now, it is almost trivial to check the first three properties, namely, that
$$d(m,n) \geq 0$$
$$d(m,n) =0 \iff m=n$$
$$d(m,n)=d(n,m)$$
and the only laborious was to check the last property (the triangle inequality). I proceeded as follows:
Let $a,b,c$ be integers, and let
$$a-b=p^s \cdot k$$
$$b-c=p^r \cdot l$$
where $l,k$ aren't divisible by $p$.
Then $$a-c=(a-b)+(b-c)=p^s \cdot k+p^r \cdot l$$
Now we have three cases, $s>r$, $r>s$ and $r=s$. We have respectively:
$$a-c=(a-b)+(b-c)=p^r  \cdot(p^{s-r} \cdot k+  l)=p^r \cdot Q$$
$$a-c=(a-b)+(b-c)=p^{s} \cdot(   k+p^{r-s} \cdot l)=p^s \cdot R$$
$$a-c=(a-b)+(b-c)=p^s \cdot (k+l)=p^s \cdot T$$
In any case,
$$d\left( {a,c} \right) \leqslant d\left( {a,b} \right) + d\left( {b,c} \right)$$
since
$$\eqalign{
  & \frac{1}{{{p^r}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}}  \cr 
  & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}}  \cr 
  & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}} \cr} $$
It might also be the case $k+l=p^u$ for some $u$ so that  the last inequality is
$$\frac{1}{{{p^{s + u}}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}}$$
$(1)$ Am I missing something in the above? The author asks to prove that in fact, if $t=t_p(m,n)$ is the exponent of $p$, that
$$t\left( {a,c} \right) \geqslant \min \left\{ {t\left( {a,b} \right),t\left( {b,c} \right)} \right\}$$
That seems to follow from the above arguement, since if $s \neq r$ then
$$t\left( {a,c} \right) = t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) = t\left( {b,c} \right)$$
and if $s=r$ then
$$t\left( {a,c} \right) \geqslant t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) \geqslant t\left( {b,c} \right)$$
$(2)$ Is there any further reading you can suggest on $p$-adicity?

REPLY [4 votes]: Your argument looks fine to me.

A very good online introduction to $p$-adics are these notes; it covers modular arithmetic leading up to Hensel's, basic analysis with the numbers, the very strange metric topology of $\Bbb Q_p$ (every point inside of a ball is a center, there are locally but not globally constant functions, etc.), and a bit of field theory and algebraic number theory. 
I don't really have any recommendation for number-theoretically in-depth stuff. Also, I tend to stick to online documents because the web is where I spend all my time anyway.
In actuality, my favorite discussion is p-adic integration and the theory of groups, which involves category theory and abstract algebra (the $p$-adics are constructed as an inverse limit of topological rings, for example), as well as measure theory and group theory. This source perhaps takes more background to digest its contents satisfactorily, but it hits my buttons well.
Just for fun, I also suggest Pictures of Ultrametric Spaces. Related entertainment: (working with adeles) the character group of $\Bbb Q$ and (working with profinite integers) profinite Fibonacci numbers.