In the book *Markov processes and Potential Theory* of Blumenthal and Getoor we can find the following result:

I don't understand the significance of this result. If I don't misinterpret the assertion, the claim is that for allmost all $\omega\in\Omega$ and for all $t\in[0,\zeta(\omega))$, the set $\{X_s(\omega):s\in[0,t]\}$ is bounded.

However, it is a basic fact that every function $x:[a,b]\to E_\Delta$ which has left and right limits at every point is bounded. So, it seems like the assertion immediately follows.

It would clearly be a stronger statement if the claim would be that $\{X_s(\omega):s\in[0,\zeta(\omega))\text{ and }\omega\in\Omega\setminus N\}$ is bounded for some null set $N\subseteq\Omega$. But that doesn't seem to be claim and it doesn't seem to be the thing which is shown in the proof (since $n$ in the last paragraph depends on $\omega$).

What am I missing? It seems like a very basic fact is proven in a complicated way.