Show that $m_{ii}=\infty$ when $i$ is transient, where $m_{ii}$ is the mean time to get from $i$ to $i$.
if $i$ is transient I know that there is a positive probability of going to some intermidiate state $j$ and never returning back to $i$. So it makes sence that after we leave we will never return again, and the mean time will be $\infty$.
However I'm strugling to prove it using formulas.
This also means that when $m_{ii}=\infty \Rightarrow$ the state can be both, null recurrent and transient.
Definitions:
$m_{ii} = \mathbb E(T_i|X_0 = i)$ and $T_i=(\min n| X_n=i$)
In words, $T_i$ is the minimum number of steps until we reach state $i$, $m_{ii}$ is the mean time it takes to go from state $i$ to $i$, in our case the Chain is Discrete, so I'm assuming time is $1$ per step
maybe this helps. page 153 ff.
https://www.math.ucdavis.edu/~gravner/MAT135B/materials/ch13.pdf
You will also find the definitions for the transition rates there by summing over probabilities P[X_n=j|X_0=i] and the expected mean time of recurrence.
So if some probabilities P[X_n=j|x0=i] of returning back to i in n steps are <1, by definition you should get a sum that has an upper bound which does not converge to any real number any more.