A $0-2$ law for cosine families with $\mathrm{lim\hspace{0.17em}sup}$ to $\mathrm{\infty }$

For ${(C(t))}_{t\in \mathbb{R}}$ being a cosine family on a unital normed algebra, we show that the estimate $\limsup_{t\to\infty^{+}}|C(t) - I| <2$impliesthat$C(t)=I$forall$t\in\mathbb R$. This generalizes the result that ${\sup }_{t\ge 0}\mathrm{\parallel }C(t)-I\mathrm{\parallel }<2$ yields that $C(t)=I$ for all $t\ge 0$. We also state the corresponding result for discrete cosine families and for semigroups.