I want to prove the following.
We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t.
(1) $f(mn) = f(m)f(n)$
(2) $f(m+n) \leq f(m) + f(n)$
(3) $0 \leq f(x) \leq 1$
then $f(m+n) \leq \max\big(f(m), f(n)\big)$?
I tried to use $f(m+n)^k$ and use (1)-(3), but I could not show the inequality.
Please help me.