Proof of a maximal inequality

Question

I am reading on martingale theory from "Probability: a graduate course" and where the following theorem is proved (in the proof, theorem $9.1$ refers to Doob's maximal inequality):

Do you have any idea how he obtained inequality $(1),$ especially how he obtained $\{\max_{1 \leq k \leq n} c_kX_{k}^+>\lambda\}$ in the integral in inequality $(1)$ ?

Answer 1

Because of the following two things:

1. $\{c_k\}$ is not increasing so the integrand is less or equal than zero.
2. $Y_k \geq c_k X^+_k$ so $\{\max_{0 \leq k \leq n} c_k X^+_k > \lambda\} \subset \{\max_{0 \leq k \leq n} Y_k > \lambda\}$.