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Let $\mathcal{A}\subset B(\mathbb{R}^n,\mathbb{R}^n)$ be a bounded family of linear maps (matrices). Now let $X_0 = \{ x\in\mathbb{R}^n ~\vert~ \lVert x \rVert \leq 1 \}$, where $\lVert \cdot \rVert$ is the usual norm on $\mathbb{R}^n$, and define $X_k$ for each $k \geq 0$ recursively as: $$ X_{k+1} = AX_k,\qquad A\in\mathcal{A}, $$ i.e. each $A$ is selected arbitrarily from $\mathcal{A}$ on each $k$.

Now suppose that there exists a finite $\varepsilon > 0$, such that $$\max_{x\in X_k}\{\lVert x \rVert\} \leq \varepsilon,\text{ for all } k,$$ i.e. all the possible sequences $(x_k\in X_k)_{k=0}^\infty$ are bounded, for any linear map combination.

Can we then conclude that there exists an $N<\infty$, such that: $$\bigcup_{k=0}^\infty X_k= \bigcup_{k=0}^N X_k.$$

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