For a Markov chain $X\_n$ I want to prove that if
$$ \mathbb{E} (\min\{n \geq 1:X\_n = i\}\mid X\_0 = i) = \infty$$
(that is, if $i$ is transient or null recurrent), then
$$ p\_{ii}^{(n)} \equiv \mathbb{P}(X\_n = i \mid X\_0 = i) \to 0 \quad \mathrm{as} \quad n \to \infty .$$
My intuition is that if the above expected value is infinite, then the probability that the first return to $i$ occurs at $n$ must decay relatively slowly with $n$, which means it's 'fairly likely' that the chain won't have returned to $i$ in the first $n$ steps. Hence the return probabilities are small. However, I can't turn this into a proof that they decay to zero. Please be as elementary as possible!

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If $i$ is transient then the theorem is easy to prove. Transience is equivalent to the sum of the $p\_{ii}^{(n)}$ being finite, which means they must tend to zero. The null recurrent case is proving more stubborn.