  For a compact group $\mathbb{G}$ acting continuously on a Banach Lie group
$\mathbb{U}$, we prove that maps $\mathbb{G}\to \mathbb{U}$ close to being
1-cocycles for the action can be deformed analytically into actual 1-cocycles.
This recovers Hyers-Ulam stability results of Grove-Karcher-Ruh (trivial
$\mathbb{G}$-action, compact Lie $\mathbb{G}$ and $\mathbb{U}$) and de la
Harpe-Karoubi (trivial $\mathbb{G}$-action, $\mathbb{U}:=$invertible elements
of a Banach algebra). The obvious analogues for higher cocycles also hold for
abelian $\mathbb{U}$.
