Let $Z^n=\{Z^n(s)\}_{s\geq 0}$, $n\in\mathbb{N}$, and $Z=\{Z(s)\}_{s\geq 0}$ be Feller processes on a complete separable metric space $V$ ($\mathbb{R}^d$ for example).
Let $D[0,t]$ the Skorohod space of cadlag functions from $[0,t]$ to $V$.
Assume we know that, for any $0<t<\bar{t}$, $Z^n$ converges weakly to $Z$ in $D[0,t]$.
Since the processes are Feller, thus strongly continuous, I would guess that then $Z^n$ converges weakly to $Z$ in $D[0,\bar{t}]$. Is this true? I don't know how to prove this rigorously.