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 

Since $i$ is recurrent, the probability of ever returning there is 1. Conditioning on the state after $k$ steps of the Markov Chain gives
$$1=P(\text{ever return to }x)=\sum_lP(\text{ever from }l\text{ to }x)p_{il}(k).$$
Now, $p_{ij}(k)>0$. So since the $p_{il}(k)$ sum to one, and  $P(\text{ever from }j\text{ to }x)$ is smaller or equal to 1, this equality shows that $P(\text{ever from }j\text{ to }x)$ must be one for the equality to hold. Therefore, an $m$ must exist such that $p_{ji}(m)>0$.
