I am stuck on the following proof that I found in Dellacherie-Meyer's book "Probabilities and potential B", p. 119 (increasing processes and projectors).
Given a map $a$ on $[0,\infty [$ which is positive, non decreasing and right continuous, they define another non decreasing map $c$ by $c(s):=inf\ \{\ t |\ a(t)>s \}$ $\forall s\geq0$.
Now they claim that this $c$ is right continuous, and I don't understand their argument. Namely, they explain that the reason is that $\{a(t)>s\}$ is the union of the sets $\{a(t)>s+\epsilon\}$ for $\epsilon>0$ (I understand that last claim).
For me, right continuity is defined by the fact that a sequence decreasing and converging to a point is sent to a sequence converging to the image of the point.
I know that there a question of this kind on stackexchange (How to prove this is right continuous?), but it doesn't seem (to me) to have much to do with the argument of Dellacherie-Meyer, and also it's about a slightly different case and I don't see how to adapt the arguments for my case.
Thanks for any help
