From the book, Probability and Measure, Billingsley, 3rd, I have problems understanding Eq. 6.6, Page 88:
$$P(\sup_k N_k \leq x ) \leq P(N_n \leq x)$$
where $N_n=1_{A_n}$ and $A_n$ is an iid sequence of events.
More specifically, I have problems with $(\sup_k N_k \leq x )$. I understand that it can be written as $$\{\sup_k N_k \leq x\} = \{\omega : \sup_k N_k(\omega) \leq x\},$$ and I know the definition $$\lim \sup_n (A_n \leq x) = \cap_{n=1}^\infty \cup_{k=n}^\infty (A_k \leq x ),$$ but I don't know how to interpret $(\sup_k N_k \leq x )$.
Thanks in advance!