Problem 2 Exercise 4.3 krieszig function analysis

Question

I am self studying Functional analysis from Krieszig and could not solve this problem

Question is:

If $p$ is a real-valued functional as defined in Hahn-Banach theorem i.e., $p$ is subadditive,

and $p( ax) = |a|\;p(x)$ , $a$ is scalar,

then prove that $|p(x) - p(y)|$ $\leqq $ $p(x-y)$ .

Please help

Answer 1

I assume that $p\ge 0$ and $p(ax) = |a|p(x)$. Then $$ p(x)-p(y) = p(x-y+y)-p(y)\le p(x-y)+p(y)-p(y) = p(x-y). $$ Similarly, $$ p(y)-p(x)\le p(y-x) = p((-1)(x-y)) = p(x-y). $$