Suppose $p(i,j)$ is a transition kernel on $S$ for a countable state markov chain $X_n$ with $$p^n(i,j)\rightarrow \pi(j)$$ as $n\rightarrow\infty$ for all $i,j\in S$. want to verify that $\pi$ is a stationary measure with respect to $p$.
We need to verify that $$\sum_i \pi(i)p(i,j)=p(j).$$
I am having trouble showing this directly. I have an inequality, but am not sure how to get equality.
$$\sum_i \pi(i)p(i,j) =\sum_i \lim_{n\rightarrow\infty} p^n(k,i)p(i,j)\leq\liminf_n\sum_i p^n(k,i)p(i,j)=\liminf_n p^{n+1}(k,j)=\pi(j)$$
Where the inequality is by Fatou, and the second last equality is because of Chapman-Kolmogorov. Any help would be appreciated.