>I think you’re right: it looks like the wiki article quotes Theorem 1.7 of the book, omitting the word stationary. I’m still a bit confused though…

Yes it looks as if the wiki article erranously quotes Theorem 1.7 which is identical to Theorem 1.8, apart that it refers to **continuous time** Markov chains, here called Markov processes, which seemed to have been of interest here in the discussion. But as I don't understand the proof on a first glance I don't want to exclude the possibility that the wikipedia author(s) did this omittance on purpose, like becasue they think the extra condition "sum to infinity" is unnecessary where of course it is also possible this is a typo or that for example the theorem was cited once correctly and that some non-mathematician erased the word "stationary because he/she eventually thought for example that the word "stationary" "sounds ugly" in bold disregard that mathematical language is less redundandant in theorem statements and in disregard that this erasure kills the whole mathematical meaning.

Anyways as said I was just more or less accidentally drawn into this discussion here and noticed this omittance in the Wikipedia article and mentioned this in order to prevent more harm in your chain of reasoning. If you think the Wikipedia article is wrong you should may be try to straighten it out.