Assume an increasing rightcontinuous $(X_t)_{t\geq 0}$ has the compensator $(A_t)_{t\geq 0}$. As saz pointed out, we want to assume that $A_t$ is continuous. Define the stopping time $\tau_s:=\inf\{t\geq s: A_t\geq A_s+\epsilon\}$ for a given $\epsilon>0$. I want to show, that this stopping time is predictable. This means on $\tau_s>0$, there exists a sequence of stopping times $\tau_n<\tau_s$ with $\lim \tau_n=\tau_s$.
Is $\tau_n:=\inf\{t\geq s:A_t+\frac{1}{n}\geq A_s+\epsilon\}$ a legit sequence?