Skorohod convergence (space of right continuous functions with left limit)

Question

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a $t \in [0,\infty)$ such that $\sup_n||f_n(t)||=+\infty$? I think the answer is no and I also think the proof shouldn't be so complicated, but for some reason I'm struggling.

Answer 1

Let us fix $t$. As $f$ is cadlag, it is bounded around $t$: $$ \sup_{|t-s|\le t/2} |f(s)| \le M $$ I use wiki notations.

Now for $n$ large enough, $$ \inf_\lambda \max \left[ \sup_{|t-s|\le t/2} |\lambda(s) - s|, \sup_{|t-s|\le t/2}|f_n(\lambda(s)) - f(s)| \right]\le \epsilon $$so there is a certain $\lambda_\epsilon$ such as $$ {|t-s|\le t/2}\implies |\lambda_\epsilon(s) - s|\le 2\epsilon, |f_n(\lambda_\epsilon(s)) - f(s)| \le 2\epsilon $$

Now let $t_\epsilon = \lambda_\epsilon^{-1}(t)\in \lambda_\epsilon^{-1} ([t\pm 2\epsilon]) \subset [t\pm t/2]$ for $\epsilon$ small enough.

Then $$ |f_n(t)| \le |f_n(\lambda_\epsilon(t_\epsilon)) - f(t_\epsilon)| + |f(t_\epsilon)| \le M +2\epsilon \le 2M $$