Prove that $i$ is an accesible state in a markov chain Let $i$ be a recurrent state of an homogeneous markov chain such that the state $j$ is accesible from $i$ (that is $\exists$ $k\ge 1$ such that $p_{ij}(k)>0$) 
Prove that $i$ is accesible from $j$ i.e. $\exists$ $m\ge 1$ such that $p_{ji}(m)>0$
Intuitively I can see why this result holds but I haven´t been able to prove it.
I tried to do it by contradiction:
suppose that $i$ is not accesible from $j$ i.e. $\forall m\ge 1$ $p_{ji}(m)=0$ 
One definition of recurrent state is that $\sum_{n=1}^{\infty}p_{ii}(n)=\infty$ 
By chapman-kolmogorov $p_{ii}(n)\ge p_{ij}(k)p_{ji}(n-k)$ I don´t know how can I procced with this
I would really appreaciate if you can help me wiht this problem 

If $i$ is not accessible from $j$, then since $p^{k}_{i,j}>0$ for some $k$, there is a nonzero probability that, starting from $i$, we will (end up in state $j$ thus) never return to $i$. This is exactly the definition of a transient state, which is by definition the opposite of a recurrent state.
See https://en.wikipedia.org/wiki/Markov_chain#Transience
