How can I prove $$\|AB\|_p ≤ \|A\|_p \|B\|_p$$ where $$\|A\|_p = \max_{x≠0}\frac{\|Ax\|_p}{\|x\|_p}, \qquad p = 1, 2, 3, \ldots$$ and known information below:
(i) $\|A\| ≥ 0$, with equality iff $A = 0.$
(ii) $\|cA\| = |c| \|A\|$, for any $c ∈ \mathbb R.$
(iii) $\|A + B\| ≤ \|A\| + \|B\|$.
*Not duplicate of Proof of matrix norm property: submultiplicativity, I don't know whats the connection between that answer an my question.