First-return probability is bounded by return probability; recurrence forces divergence

Suppose $f_{ii}^{(n)}$ denotes the probability of the first return to $i$ in $n$ steps. (I) For any state $i$, $f_{ii}^{(n)}\le p_{ii}^{(n)}, n\ge1$. (II) For any state $i$, $\sum_{n\ge1}p_{ii}^{(n)}\ge1$. (III) For a persistent state $i$, $\sum p_{ii}^{(n)}\ge1$.