Is there a sequence of $ C^\infty $ compactly supported functions limiting to $ f(x)$=1

Question

Is there a sequence $\left(g_m\right)_{m\geqslant 1}$ of compactly supported $C^{\infty}$ functions converging to $ f(x)=1$ uniformly with the condition that there is no compact $K \subset \mathbb R$ such that
$\operatorname{card}\left\{ m: \operatorname{supp}(g_m)\cap K \ne \emptyset\right\} = \infty$?

Answer 1

There is no such function even with pointwise convergence.

Hint: choose $K=\{0\}$ (a finite, hence clearly compact, set).