Assume $X=(X_t)_{t \in [0,\infty)}$ is a RCLL adapted process. I set $X_{-t}:=\lim_{s \rightarrow t, s < t}X_s$ for $t \in (0,\infty)$ and $X_{-0}=0$. I want to show that $(X_{-t})_{t \in [0,\infty)}$ is locally bounded. I tried defining a stopping time by setting
$$\tau_n := \inf\{t \in [0,\infty) : |X_t| \geq n \}.$$
Then $(\tau_n)_{n \in \mathbb{N}}$ is a stopping time by adaptedness and the RCLL property of $X$. And also for every $\omega \in \Omega$, we have for $t>0$ that
\begin{align} |X_{-t}^{\tau_n(\omega)}(\omega)1_{\{\tau_n > 0\}}(\omega)|&=|X_{-t}^{\tau_n(\omega)}(\omega)|1_{\{\tau_n > 0\}}(\omega) \\\\ &= |\lim_{s \rightarrow t \wedge \tau_n(\omega), s < t \wedge \tau_n(\omega)}X_s(\omega) |1_{\{\tau_n > 0\}}\\\\ &=\lim_{s \rightarrow t \wedge \tau_n(\omega), s < t \wedge \tau_n(\omega)}|X_s(\omega) |1_{\{\tau_n > 0\}} \\\\ &\leq n \end{align}
And clearly by definition, this also holds for $t=0$, since we set it to $0$ there. Now whats missing, is to show that $\tau_n(\omega) \rightarrow \infty$ for all $\omega \in \Omega$.
But I do not see how I can deduce that. Can someone help? Is the approach with my stopping time even the right one?
Thanks a lot?