norm of a product of matrices Suppose for every $x \in \mathbb{R}$ and $y \in [0,1]$, $M(x,y)$ is an $n$ by $n$ matrix  and suppose that for every $y \in [0,1]$, $M(x,y) \to M_\infty$ as $|x| \to \infty$, where $M_\infty$ is a constant matrix (where the norm is the operator norm considering the matrix as a linear operator). 
Also, suppose that spectral radius of $M_\infty > 1$. (i.e we also have that $||M_\infty|| >1$)
Define the product $M_n(x,y) $ as 
$M_n(x,y)=M(x,y)M(x+y,y) M(x+2y,y) \ldots M(x+(n-1)y,y)$. Would it be possible to prove something like 
$$ \lim_{n \to \infty} \bigg(\mbox{sup} ||M_n(x,y)||\bigg)^{1/n} \geq ||M_\infty||$$
where the supremum is taken over all $x \in \mathbb{R}$ and all $y \in [0,1]$. 
Thank you. 
EDIT: What I was looking for is an inequality of the form 
$$ \lim_{n \to \infty} \bigg(\mbox{sup} ||M_n(x,y)||\bigg)^{1/n} \geq spr(M_\infty) $$ which is completely answered by Martin Argerami below. 

No. Take $M(x,y)=M_\infty$ for all $x,y$, where $M_\infty$ is a non-zero nilpotent matrix. For large $n$, we have $M_n(x,y)=0$.
