Convergence of p_ji (k) in irreducible, aperiodic finite Markov chains

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Hello, I can't prove the blue step? Can someone see how? It is known: All terms are =>0, m_ii=Sum k=1 to inf kf_ii, f_ji=sum k=1 to inf f_ji (k)=<1 . Sum i=1 to N of p_ji(k) =1 for all k. Thank you in advance.