Let $Z$ be a (not neccesarily cadlag) process adapted to some filtration $\mathbb{F}_t$. Assume that $T_n,T$ are stopping times so that $T_n<T_{n+1}$ and $T_n\uparrow T$. Assume that $Z^{T_n}$ is a bounded continuous martingale, but almost surely $Z_t\rightarrow \infty$ for $t\rightarrow T$.
I strongly believe that the limit $[Z^{T_n}]_{T_n}\rightarrow \infty$ for $t\uparrow T$ holds . Can anybody supply with a proof or a reference?