I am looking at the following theorem and proof
For the purposes of my question, one need only know that $$p_{i,j}(n)=:P(X_n =j \mid X_0 =i)$$ I am having issues with when the author says that if $j$ is transient, then the sum is finite. The assumption seems to only give us that each therm is $<1$ but nothing more. Moreover, I would think that the assumption is just $$P(\text{Getting j on the first 1st or 2nd or 3rd or ...})$$ is less than $1$, which is cleraly smaller than the sum as per the product rule.
Could someone help me understand why is the sum finite?