@LSpice: Sure, the paper I'm reading is in this link arxiv.org/pdf/1806.00221.pdf, and the proof, or the proposition, is in page 9.

By the way, would it be considered wrong to say that, $F \to 1$ for $t \to \infty$, as $e^{-\int_{t_n}^{t}\lambda^*(s)ds} \to 0$ for $t \to \infty$? Because does it not do that?

Yemon Choi: Thank you that makes sense. I'll oplaod my question in math math.stackexchange.com.

Moreover, this proof's objective is to show that $F$ is in fact a cumulative distribution function (which thus is proofed by giving the three points). I hope this made it more clear.

I hope this is the answer you are looking for. The paper is about temporal point processes, more specifically, evolutionary temporal point processes (In other words, the present event (point) is dependent of the previous event(s)). This is a proof I'm trying to understand. I just can't seem to find the correct arguments to state why these two assumptions imply the three points.