# Union of images under arbitrary linear maps with boundness assumptions

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: $$\bigcup_{k=0}^\infty X_k= \bigcup_{k=0}^N X_k.$$