my task is to proof the following: Let $H^n$ be a sequence of uniformly bounded predictable processes such that $$\lim_{n\to\infty}H^n_t= 0$$ almost surely holds for all $t\geq 0$. Then we have $$H^n\xrightarrow{ucp} 0,$$which means $\sup_{s\in [0,T]} H^n_s\to 0$ for all $T\geq 0$.
The problem is, I am not even sure whether this statement is true. Assume the deterministic process $$ H^n=\begin{cases} t^n&\text{for } t<1\\ 0&\text{for } t\geq 1.\end{cases}$$
Then we have $$H^n_t\to 0 \text{ for all } t\geq 0$$ but $\sup_{s\in[0,1]} H^n_s = 1$ and hence $H^n\not\xrightarrow{ucp} 0$.
My ansatz for a proof would be to show the statement for continuous processes (which seems fairly easy) and then apply the monotone class theorem.
Can anyone tell me whether this statement is actually true and in case it is, what is wrong with my "counterexample"?
Thank you very much.