I am trying to solve the following question.
Let $(u_n)_{n=1}^{\infty}$ be an unconditional basis for a Banach space $X$ with suppression-unconditional constant $K_{su}$. Prove that for all $N$, whenever $a_1,...,a_N,b_1,...,b_N$ are scalars such that $|a_n| \leq |b_n|$ for all $1 \leq n \leq N$ and $a_nb_n>0$, we have
$\|\sum_{n=1}^N a_n u_n\| \leq K_{su} \|\sum_{n=1}^{N}b_n u_n\|$
If somehow I can prove the unconditional constant $K_u \leq K_{su}$ then I will be done. I don't know how to use the fact that $a_nb_n > 0$.