I'm reviewing the proof of the theorem of oseledet the book Mañe:
Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. Let $L: F \rightarrow F$ vector bundles covering $f$ (i.e. $\pi\circ L= f\circ \pi$) such that both $L$ and $L^{-1}$ have bounded normas. Denote by $L_n$ the n-th iterate of $L$ $$L_n(x)=L(f^{n-1}(x))\circ \ldots \circ L(f(x))\circ L(x) \ \ \ \mbox{if} \ \ n>0,$$ $$L_n(x)=L^{-1}(f^{-n+1}(x))\circ \ldots \circ L^{-1}(f^{-1}(x))\circ L^{-1}(x)\ \ \ \mbox{if} \ \ n<0.$$ For $n_1,\ldots, n_l\geq 1$ fixed. Let $A_k$ the set of 2l-uples of rational numbers $\alpha_1>\beta_1\ldots \alpha_l>\beta_l$ with $(\alpha_i-\beta_i)<\frac{1}{k}$ for $1\leq i\leq l.$
For $m\geq 1$ and ($\alpha_1,\ldots,\beta_l)\in A_k$, let $\Lambda(m,\alpha_1,\ldots,\beta_l)$ be the set of point $x\in M$ for which there is a splitting $F_x=F_1(x)\oplus \ldots \oplus F_l(x)$ with $dimF_j=n_j$ and $$exp(n\alpha_j)\Vert u\Vert \geq \Vert L_n(x)u\Vert \geq exp(n\beta_j)\Vert u\Vert $$ $$exp(-n\alpha_j)\Vert u\Vert \leq \Vert L_{-n}(x)u\Vert \leq exp(-n\beta_j)\Vert u\Vert $$ for all $n\geq m$, $1\leq j\leq l$ and $u\in F_j$.
Then the set $\Lambda(m,\alpha_1,\ldots,\beta_l)$ is closed and $\Lambda(m,\alpha_1,\ldots,\beta_l) \ni x \rightarrow F_j(x)$ is contionuos for every $1\leq j\leq l$. This is where I have difficulty
Let $(x_n)\subset \Lambda(m,\alpha_1,\ldots,\beta_l)$ such that $x_n \rightarrow y\in M$, then there $F_{x_n}=F_1(x_n)\oplus \ldots \oplus F_l(x_n)$ with $dimF_j(x_n)=n_j$ but as I can guarantee that $F_{x_n} \rightarrow F_{y}$? how to make the convergence of subspaces?
thanks for any suggestions