Prove: Birth-death chain has stationary distribution -> satisfy detail bal. equations

#1
I'm working my way through a course in stochastic processes but came across this Lemma in my coursebook where they didn't show the working (Lemma 8.5). I'm not familiar with doing proofs and it isn't taught in any of my courses but I can read/understand them, how did they reach the conclusion of lemma 8.5?

 
#2
Re: Prove: Birth-death chain has stationary distribution -> satisfy detail bal. equat

So I've been browsing youtube videos and ended up coming close to what i wanted but its still slightly different. What am I missing here?