Let $(\sigma_t)_{t\ge 0}$ and $(b_t)_{t\ge 0}$ be two progressively measurable $\mathbb{R}^{m \times d}$ and $\mathbb{R}^m$-valued locally bounded processes. Then I want to show that there is some $\Omega_0 \subset \Omega$ with $P(\Omega_0)=1$ and $$\max_{j,k}\sup_{t \le T} |\sigma_{jk}(t,\omega)|+\max_j \sup_{t \le T} |b_j(t,\omega)|<\infty \; \text{for all}\; T>0,\omega \in \Omega_0.$$
My approach: From the definition of locally bounded processes, we have some localizing sequences $\tau_n \uparrow \infty$ a.s., for which $\sigma^{\tau_n} 1_{\tau_n>0}$ is uniformly bounded. Since we have finitely many $j,k$ for $\sigma_{jk}$ and $b_j$, and the minimum of localizing sequences is also a localizing sequence, we can fix a single localizing sequence $\tau_n$ for all $md+m$ one-dimensional processes.
Now, take $\Omega_0$ to be the set in which $\tau_n \uparrow \infty$. Then $P(\Omega_0)=1$.
If we fix $\omega \in \Omega_0$, then since $\tau_n(\omega) \uparrow \infty$, for any $T>0$, we have some $n$ for which $\tau_n(\omega)>T$. Hence we have $\sup_{t \le T}|\sigma_{jk}(\tau_n(\omega) \wedge t,\omega)| 1_{\tau_n(\omega)>0} = \sup_{t \le T} |\sigma_{jk}(t,\omega)| \le K$.
Next, we can do the same for $\sup_{t\le T} |b_j(t,\omega)| \le K'$, then for all $j,k$ since there are only finitely many of them.
Thus we have shown that $$\max_{j,k}\sup_{t \le T} |\sigma_{jk}(t,\omega)|+\max_j \sup_{t \le T} |b_j(t,\omega)|<\infty \; \text{for all}\; T>0,\omega \in \Omega_0.$$
Is this proof correct?