Question about convergence of matrices (Linear algebra) Let $O(n)$ to be the orthogonal group, and define $m(A)$ for every $A \in O(n)$ as follows:
$$m(A) = \max \left\{ \frac{|Ax-x|}{|x|} : x \in \mathbb{R} ^ n -\{0\} \right\}\;.$$ 
Given a sequence $ \{B_i \}$ , we can assume by compactness that it converges. 
Why does this imply that for every $\epsilon>0$  there exist $i<j$ such that $m(B_j B_i^{-1} )\leq \epsilon $ ? 
Thanks in advance 

Suppose $B_i \to L$ as $i \to \infty$. Since both multiplication and inverse are continuous, as $i \to \infty$ and $j \to \infty$ we have $B_j B_i^{-1} \to L L^{-1} = I$.  Now $m$ is continuous (note that by homogeneity we may take $x$ to be in the unit sphere which is compact), and $m(I) = 0$.
