I have this following problem
Let $(X_i)_{i \in \mathbb{N}} $ be i.i.d. random variables with $X_i \geq 0$ for all $i \in \mathbb{N}$. Let $S_n=\sum_{i=1}^n X_i. \, S_0:=0$
For $k \in \mathbb{N}$, let $N_k(t,t+h):=\sum_{n \geq k} 1_{(t,t+h]}(S_{n+k}-S_k)$.
Show that
$N_0(t,t+h) \leq N_k(0,h)+1$ for all $k \in \mathbb{N}$,$t,h>0$,
on the set $A_k=\{S_{k-1} \leq t < S_k \}$.
Now this doesnt look too hard at the first glance and Ive made some progress but even after trying a long time, Im just not able to proof it. Can somebody help?