In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$

Because I'm working with a kind of sequence of shif-invariant measures such that the sequence of entropies is monotone increasing (it could be constant...) and I couldn't find in the literature some good "machinery" to prove that convergence... I've tried, for instance, the $\bar{d}$ distance related to joinings (defined here joining or coupling) but, it didn't work too well...(I mean, I was not able to work with...)

So, any tips, hints?

Thanks for your attention

kind regards