Mañé's lemma (Mañé 1983; González-Tokman and Quas 2015, Lemma 15), from the density argument: telescoping $g ≤ h + g ∘ σ$ bounds $g$ at a time $n$ by its value at a later time where it is known to be small, plus a Birkhoff sum of $h⁺$ over a window of length $o(n)$.