Triangle inequality for the $p$-adic metric

Question

I try to understand the triangle inequality prove for the $p$-adic metric. The proof is given as:

$$\DeclareMathOperator{ord}{ord}|x-y|_p = p^{-\ord_p(x+y}\leq p^{-\min \{\ord_p(x),ord_p(y)\}} = \max \{p^{-\ord_p(x)},p^{-\ord_p(y)}\} = \max \{|x|_p,|y|_p\}$$

Now I don't understand why $p^{-\min \{\ord_p(x),\ord_p(y)\}} = \max \{p^{-\ord_p(x)},p^{-\ord_p(y)}\}$ how we went from min to max?

Answer 1

It’s a consequence of the minus sign in the exponent. Look at a concrete example:

$$p^{-\min\{2,3\}}=p^{-2}=\max\{p^{-2},p^{-3}\}\;,$$

for instance. Making a positive quantity $n$ smaller makes $p^{-n}$ larger.

Answer 2

The function $n \mapsto p^{-n}$ is strictly monotonically decreasing: $$n < m \implies p^{-n} > p^{-m} $$