I want to understand theorem 5.21 (page 224) in [this link](https://www.math.wisc.edu/~roch/mdp/roch-mdp-chap5.pdf) and here is where I don't understand: $$
\{W = t\} = \{t \text{ is the first ladder index in }R\_1, \dots, R\_t\},$$
i.e. $\{R\_t = 1, R\_1 < 1, \dots, R\_{t-1} < 1\}$.


And if anyone can offer easier ways to prove it, it would be great to hear your ideas.



Proof of Hitting-time theorem in branching processes
In the proof of Theorem 5.21 (page 224) in the notes at your link, we find the definitions 
$R\_i:=1-S\_i$, with $R\_0:=0$. By Lemma 5.17 and the three-line display on page 221 of those notes, 
$$W=\tau\_0=\inf\{t\ge0\colon S\_t=0\},$$
whence 
$$W=\tau\_0=\inf\{t\ge0\colon R\_t=1\}.$$
Thus,
$$W=t\iff R\_1<1,\dots,R\_{t-1}<1,R\_t = 1,$$
as claimed. 


