Mean recurrence time versus first-return probability

The following are three statements related to a Markov Chain: (I) For any state $i$, $\mu_i\ge f_{ii}$. (II) For a persistent state $i$, $\mu_i\ge1$. (III) For a transient state $i$, $\mu_i<1$, where $\mu_i$ denotes the mean recurrence time for state $i$ and $f_{ii}$ is the probability of first return to state $i$.