Consider a random variable $\tau$:
$\tau := \inf \left\{ {k = \sigma , \ldots ,T:S_k \ge u} \right\}\wedge T,$ where $\sigma$ is a stopping time, $S_k$ - stochastic process, $T, u$ are constants and $a\wedge b=\min(a,b)$. I would like to prove that $\tau$ is a stopping time, i.e. I need to show that $\{ \tau\le k\} \in {F_k}$, where ${F_k}$ is natural filtration.
I tried to present $\tau$ as $\{ \tau \le k\} = \mathop \bigcup \limits_{j = \sigma }^k \{ \tau = j\} $ and to show that $\{ \tau = j\}\in F_j, {F_j} \subset {F_k}$, so I can conclude that $\{ \tau\le k\} \in {F_k}$. But can I really use a stopping time $\sigma$ in union operator as starting index? If not, what is the correct way to prove that $\tau$ is a stopping time?