Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ (we may assume it is countable) is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true that for any $x\in S$ $$\underset{n\rightarrow\infty}{\text{lim}}~\underset{\text{A}\in\mathcal{F}^{\mathbb{Z}_+}}{\text{sup}}~|P_x((X_{n},X_{n+1},\dots)\in\text{A})-P_\pi(X\in A)| =0?$$

I would assume this holds true due to the similarity of the statement to the usual convergence theorem $$\underset{n\rightarrow\infty}{\text{lim}}~||P_i(X_n\in\cdot)-\pi(\cdot)||_V=0$$ and the feeling that the Markov chain is reproducing itself. But I can't figure out how to prove it. I tried to come up with coupling which seems to be the way to deal with those Variance norms. Thanks a lot for any suggestion.