proof T is a stopping time

Question

let $X(t) $ is a stochastic process and is cadlag and adapted, let $T = \inf\{t:|X(t)| \ge c\}$, proof T is a stopping time. i.e.$\{T\le t\} \in F_t$

Answer 1

$\{T \leq t\}^c = \{T > t\} \subset \{|X(s)| < c , \forall s \leq t\}$

Since $X_t$ is right continuous, $|X_t| < c$ implies $\exists \epsilon >0$ such that $|X_{t+\delta}| < c , \forall \delta < \epsilon$. So in this case we have $T > t$. Therefore we see $\{|X(s)| < c , \forall s \leq t\} \subset \{T > t\}$.

So we see $\{T > t\} = \{|X(s)| < c , \forall s \leq t\} \in \mathcal{F}_t$, then its complement $\{T \leq t\} \in \mathcal{F}_t$