Solution
= Solution
In a transient irreducible chain, every state $j$ is transient, so the expected number of visits satisfies
$$
\sum_{n=0}^\infty P^n(i,j)=\mathbb E_iV_j<\infty.
$$
The terms are nonnegative, and every convergent series has terms tending to zero. Therefore
$$
\boxed{P^n(i,j)\to0}.
$$