  Let $(C(t))\in\mathbb{R}}$ be a cosine function in a unital Banach algebra.
We give a simple proof of the fact that if lim sup$\_{t\to 0}\vert
C(t)-1\_A\vert\textless{}2,$ then $lim sup\_{t\to 0}\Vert C(t)-1\_A\Vert=0.$
