A symmetric walk reflected at 0 is null recurrent

Consider a Markov chain on $S=\{0,1,2,\dots\}$. The transition probabilities are given by $p_{00}=\tfrac12$, $p_{01}=\tfrac12$ and for $i\ge1$, $p_{i,i-1}=\tfrac12$, $p_{i,i+1}=\tfrac12$. Then: