I know that for the infinite 1-D random walk, it is persistent (my professor showed us by showing that $\displaystyle \sum_{n=1}^{\infty}{P_{jj}(n)} = \infty$, so $\displaystyle f_{jj}=1$), which makes me think that that the semi-infinite walk is also persistent, but I'm having a hard time showing it. The problem I was given is below, any help would be greatly appreciated. $\displaystyle f_{ij}$ below is the probability of the first return (or visit) to state j starting in state i

A random walk is defined on the integers {0,1,2,3,...} with the following
transition probabilities:
$\displaystyle p_{00}=1/2$ , $\displaystyle p_{01}=1/2$
$\displaystyle p_{n,n+1}=1/2$ , $\displaystyle p_{n,n-1}=1/2$

Determine whether the walk is transient or persistent. [Hint: relate the walk
to the standard symmetric walk on the integers. The standard symmetric walk is persistent, thus $\displaystyle f_{0,1}=1$. By conditioning on the fi rst step of the walk starting at 0, use this to compute $\displaystyle f_{0,0}=1$ for the walk on the half-line given above].