I'm currently trying to understand the proof of the final theorem of (Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time) where I need the following: For $\{M(t)\}_{t\geq0}$ a càdlàg local martingale we define $$a^\epsilon[M](t)=\sum_{s\leq t}|\Delta M(s)| 1\{|\Delta M(s)|>\epsilon\}$$ and denote it's compensator by $â^\epsilon[M]$.

We then define $\overline{M}^\epsilon:=a^\epsilon[M]-â^\epsilon[M]$ and $\underline{M}^\epsilon:=M-\overline{M}^\epsilon$.

"It can be shown that the maximal jump in $\underline{M}^\epsilon$ has size less or equal than $2\epsilon$."

Why is that? I also tried looking up in the references but ended up with "LE THEOREME FONDAMENTAL SUR LES MARTINGALES LOCALES" by Meyer which says something about "totally inaccessible times", on which I couldn't find enough to make sense of the proof (which is written in French, which I don't speak…)

Thanks a lot in advance.