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I am looking at the following theorem and proof

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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?

Maths Wizzard
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  • I think knowing what $P_{I,j}(s)$ represents is also highly relevant information for answering this question. – Paul Sinclair Aug 25 '22 at 20:15
  • Why should it matter? I am only asking why is the sum finite. Regardless, here is the definition $P_{i,j}(s)=\sum_{n=0}^{\infty}p_{i,j}(n)s^n$ – Maths Wizzard Aug 25 '22 at 22:11
  • Because the whole argument is about it being finite. Reasons for it to be finite might be found in what it represents. For example, if $P_{ij}(s)$ had been itself a probability, then obviously it would have to be finite. Without knowing what it is, one cannot explore those possibilities. Even as a generating function defined from the $p_{i,j}$, you obviously have already determined facts about it, as indicated by the "Recall that". The proof if its finiteness when $i = j$ may lie in those previous facts. – Paul Sinclair Aug 26 '22 at 02:28

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