In fact, what I really want to prove is the equality. The another inequality is already proven, but this one I'm not able to prove. Fix an invertible matrix $A\in\mathbb{R}^{n\times n}$ and a vector $c\in\mathbb{R}^n$, I need to prove the following inequality.
$$\max_{\|M\|_{rs}\leq 1,\ \|b\|_s\leq 1} \|A^{-1}(Mc-b) \|_r \geq \|A^{-1}\|_{sr}(\|c\|_r+1) $$
$r,s$ are positive integers and $\|\cdot\|_{rs}$ is the matrix norm induced by the norms $\|\cdot\|_r$ and $\|\cdot\|_s$.
Thank you very much for your help.
EDIT: Ok, I have an answer and I'm very grateful for it. But what I should have said is that I want a direct proof. Is it possible to exhibit $M$ and $b$ in order to get an equality?
Suppose that, for all $ \ M \in \overline{B_{rs} ( 0 , 1 )} \ $ and for all $ \ b \in \overline{B_s (0,1)} \ $, we have that $$|A^{-1}| \cdot (|c| + 1) < | A^{-1} (Mc - b) | \ . $$ So, for all $ \ M \in \overline{B_{rs} ( 0 , 1 )} \ $ and for all $ \ b \in \overline{B_s (0,1)} \ , $ one has $$|A^{-1}| \cdot (|c| + 1) < | A^{-1} (Mc - b) | \leqslant |A^{-1}| \cdot |Mc - b| =$$ $$= |A^{-1}| \cdot |Mc + (-b)| \leqslant |A^{-1}| \cdot (|Mc| + |-b|) = |A^{-1}| \cdot (|Mc| + |b|) \ . $$ Take $ \ M= id_{\mathbb{R}^{n \times n}} \ \ $ and $ \ b = (1,0,0,...,0) \ $ to get a contradiction.