For a discrete-time Markov chain that is not necessarily irreducible or aperiodic, I am attempting to show that for transient $j$ \begin{equation*} \lim_{n\to\infty}p_{ij}^{(n)} = 0. \end{equation*} There is a well known result that states if \begin{equation*} \sum_{n=0}^{\infty}a_{n}<\infty, \end{equation*} then $a_{n}\to 0$. Since the $p_{ij}^{(n)}$ are a sequence of numbers, if we use that for transient $j$ \begin{equation*} \sum_{n=0}^{\infty}p_{ij}^{(n)} <\infty \end{equation*} is it possible to deduce that \begin{equation*} \lim_{n\to\infty}p_{ij}^{(n)} = 0? \end{equation*}
I am slightly doubtful in the above statement as there is another well known result (see, e.g., pg. 127 Theorem 8.8 in Billingsley's Probability and Measure 2nd Ed.) that states $p_{ij}^{(n)}\to 0$ for all $i$, $j$ in an irreducible, aperiodic chain which is transient. Another post, Prove that the limiting probability of a transient state in a Discrete Time Markov Chain, is 0, appears to have asked the same question but the only answer to the question refers to the previously mentioned result for irreducible, aperiodic chains. However, the result that I am trying to prove seems intuitive so I am having trouble understanding why it would not be true. My first thought is that maybe there is a misunderstanding in what is being asked in the question linked, as there appears to be possibly a slight difference in quantifiers unless I am misunderstanding. My second thought is that maybe the question in the provided link should be phrased such that $p_{jj}^{(n)}\to 0$ for transient state $j$, since we know that \begin{equation*} \sum_{n\geq 0}p_{jj}^{(n)} < \infty \end{equation*} for transient $j$.
I am primarily looking to see if anyone can confirm if the approach above would work to prove the statement, or if anyone can provide a counterexample using the fact that the chain must be irreducible and aperiodic to show that this is a necessary condition for $p_{ij}^{(n)}\to 0$ for transient $j$.
** Edit: definition of transient state is \begin{equation*} f_{ii} = P(X_{n} =i,\ n\geq 1\mid X_{0}=i) < 1. \end{equation*}