Consider a continuous local martingale $\{X_t\}_{t\geq 0}$ with $X_0=0$, and $\limsup_{t\to\infty}X_t = \infty$ and $\liminf_{t\to\infty}X_t = -\infty$ with probability one. Then, I want to show that we can embed a simple random walk into $\{X_t\}$. That is, if we define recursively $\tau_n:=\inf\{t>\tau_{n-1}:|X_t-X_{\tau_{n-1}}|=1\}$ with $\tau_0=0$, then $\{X_{\tau_n}\}_{n\in \mathbb{N}}$ is a simple random walk.
I use the continuity of $\{X_t\}$ and the fact that its the range is $\mathbb{R}$ almost surely to deduce that $\mathbb{P}(\tau_n<\infty)=1$ for all $n$. Then, we have $X_{t\wedge \tau_n}\to X_{\tau_n}$ as $n\to\infty$ a.s. and, by the bounded convergence theorem, we have $0=\mathbb{E}X_{t\wedge \tau_n}\to\mathbb{E}X_{\tau_n}$ for all $n$ as $t\to\infty$. With this, I was able to show that $\mathbb{P}(X_{\tau_1}=1)=\mathbb{P}(X_{\tau_1}=-1)=1/2$, but couldn't figure out how to continue. I'd appreciate any help, thank you!