Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), W^u_\delta(x)$ for sufficiently small $\delta>0$. its well known that $W^s(x)=\cup_{n\geq0}f^{-n}(W^s_\delta(f^n(x))) , W^u(x)=\cup_{n\geq0}f^{n}(W^u_\delta(f^{-n}(x)))$ are absolutely continuous foliations on $M$.

An argument which i have met several times is that $\cup_{x\in W^u_\delta(y)}W^s_\delta(x)$ contains an open ball $B(y, r)$ for some $r>0$.I can see this in a geometric interpretation but i can't proof it. i was wondering if someone could help me please.