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I'm actually working through the Renewal Limit Theorem Proof from "Einführung in die Wahrscheinlichkeitstheorie und Statistik" by Ulrich Krengel and having problems to understand the first step, everything else is fine.

He defines: $f_m := f_{jj}^{(m)} = $ Probability of reaching state j for the first time again after time m

$p_{jj}^{(n)} := $ Probability of reaching state j again after time n

$u_n := p_{jj}^{(n)} = \sum_{m=1}^n {f_m u_{n-m}}$

$\lambda := \limsup u_{n_k}$

For a sequence $n_k \rightarrow \infty$ and $u_{n_k} \rightarrow \lambda$ and for every $m \geq 1$ there is: $$\lambda = \lim_{k\rightarrow \infty} u_{n_k} = \lim_k(f_m u_{n_{k-m}}+\sum_{1\leq s\leq n_k, s\neq m}{f_s u_{n_{k-s}}}) \leq \liminf_k(f_m u_{n_{k-m}}) +\limsup_k (\sum_{1\leq s\leq n_k, s\neq m}{f_s u_{n_{k-s}}})$$

I don't really understand why he sets $\lambda = \limsup u_{n_k} = \lim_k u_{n_k}$ and later uses a sum of $\limsup$ and $\liminf$ which is $\leq$ than the $\lim$ itself.

Any help would be greatly appreciated.

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