Criterium for recurrence and Borel-Cantelli lemma

Question

Given a Markov chain $X$ (discrete time, countable state space) we know that a state $i$ is recurrent iff $$\sum_{n\geq1}p_{ii}^{(n)}=\infty$$ where $p_{ii}^{(n)}=P(X_n=i|X_0=i)$. Otherwise, $i$ is said to be transient. I would like to prove this using Borel-Cantelli lemmas. I found here (lemma 2.1) that Borel-Cantelli convergence part still holds for conditional probabilities so my idea is $$\sum_{n\geq1}p_{ii}^{(n)}<\infty\text{, by Borel-Cantelli}\implies\sum_{n\geq1}\mathbb{1}_{\{X_n=i\}}<\infty\quad a.s.$$ so we get just a finite number of visits to $i$ and therefore $i$ is transient. Is there anything wrong with this reasoning? I ask because I cannot find any proof of this necessary and sufficient condition, based on Borel-Cantelli. Also, I would appreciate any suggestion on how to prove that $i$ transient implies the convergence of the series.