Time parametrization for Brownian Motion

Question

Let $(B(t))_{t \geq 0}$ be a Brownian Motion. Define following Stopping times $T_{k,0}=0 $ for every k and $T_{n,k}= \inf \{ t>T_{n,k-1} \vert \vert B(t)-B(T_{n,k-1}) \vert \geq 2^{-n} \} $. By definition $T_{n,k}< \infty $ for all n and k. Then for every $n,k$ and $\tilde{n}$ there exist a $\tilde{k}$ such that $T_{n,k}=T_{\tilde{n},\tilde{k}}. $ We write $\tau_{\tilde{n}}(n,k)$ for this $\tilde{k}$. It is possible to show that $$ \lim_{m \rightarrow \infty} 4^{-m}\tau_m(n,k) $$ exist for every n and k. Define now $$\tau(n,k)= \lim_{m \rightarrow \infty} 4^{-m}\tau_m(n,k)$$ and $$ \tau(t)=\sup \{ \tau(n,k) \vert T_{n,k} \leq t \} $$. Now i want to show that for every t we have almost surley $$ \tau(t)=t $$. I would really appreciate some help for this question. Already thanks!