They gave a statement saying so. Whatever the case, there's no 1:1 relationship between show viewership and profit in any traditional sense, only what they consider to be successful using their own internal metrics. And their execs outright stated it to be internally successful and profitable, saying themselves it made back their money.
https://collider.com/rings-of-power-...ders-comments/
"As big of an investment as it's been, it has more than paid off for us."
This would be the proof. If there is any skepticism, there would need to be proof to show this to be a lie.