Lemma 1

Let U be an open subset of $\mathbf{C}^n$ and K a compact subset of U. There exists an infinitely differentiable mapping $h$ of $\mathbf{C}^n$ into $\mathbf{C}$, equal to 1 on K and with compact support contained in U.

Let V be a relatively compact open neighbourhood of K such that $\overline{V}$ is contained in U (TG, I, p. 65, prop. 10). There exists an infinitely differentiable function $h$ of $\mathbf{C}^n$ into $\mathbf{C}$ whose support is contained in V and which is equal to 1 on K (VAR, R1, p. 40, 5.3.6). This function has the required properties.