###### Proof: Let $i\in C$, such that $i$ is recurrent. Then $f_{i}=\mathbb{P}(T_{i}<\infty \mid X_{0} = i)=1$) i.e. $\sum_{n=0}^\infty P^n_{ii} = \infty$Let $j \in C\setminus\{ 0 \}$ then there exist $a,b\in \mathbb{N}$ sucht that $P_{ij}^a <$