I'm trying to understand the Ergodic Theorem, but the book doesn't explain a step in this calculation. I understand the first step, and I understand the reason for why it converges to $0$, but what I don't understand is why this is equal to $\frac{1}{m_i}$.
Is it enough with just a simple intuitive explanation of this. The parameters are:
$V_i(n)$ = Visits in $i$ before $n$
$V_i$ = Total number of visits before $i$
$m_i = E_i(T_i)$ = Expected time to return to state $i$
$\frac{V_i(n)}{n} < \frac{Vi}{n} \to 0 = \frac{1}{m_i}$