Assume we have a sequence, $(O_n)$, of bounded linear operators which map from $X$ to $X$ a finite dimensional Banach space. Also assume each operators has the same spectral radius, and $\rho(O_1)<1$.
Prove that $\lim_{n\to \infty}O_nO_{n-1}\cdots O_{1}=$ Null operator.
It is not true in general. Take $O_{2k+1}=O_1=\begin{bmatrix}0&1\\0&0\end{bmatrix}$ and $O_{2k}=O_1^T$.
If the operators were commuting and there were a uniform bound on the norms (for any fixed operator norm), I think it would be true. Just commuting is not enough, as examples of the form $\begin{bmatrix}\lambda&a_n\\0&\lambda\end{bmatrix}$ show, with $0<|\lambda|<1$ and $|a_n|$ growing rapidly enough. The above example shows having a uniform bound isn't enough.