A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Question

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued continuous functions with domain $X$. I want to show that if $f(x)=0$, then $h(x)=0, \forall h \in I$.

I need this to prove $V(I):=\{x \in X : h(x)=0, \forall h \in I\}$ is open in $X$.

Please help. Thanks in advance

Answer 1

$f(x) = 0$ implies $f_m(x)=0$ for all $m$, since $\sum_{m} |f_m(x)|$ is a sum of non-negative numbers, which is zero.