# The intersection of zero loci of a family real analytic functions

If $$(f_i)_{i in I}$$ is a family of holomorphic functions on some open set $$U subset mathbb C^n$$ with zero loci $$V_i$$, then the intersection $$bigcap_{i in I} V_i$$ is complex analytic. Moreover, for every compact $$K subset U$$ there exists a finite subset $$J subset I$$ such that
$$K cap bigcap_{i in I} V_i = K cap bigcap_{i in J} V_i ,.$$
See Is intersection of zero set of any family of holomorphic functions an analytic set?

Now take the same setup but replace “analytic” by “real analytic” and $$mathbb C$$ by $$mathbb R$$.

The first statement is then trivially true, or at least if $$I$$ is countable: define $$f = sum_{i in I} c_i f_i^2$$ for very rapidly decreasing $$c_i>0$$, then the zero locus of $$f$$ (where it converges) is $$bigcap_{i in I} V_i$$.

Is the second statement also true? That is:

Does there exist for every compact $$K subset U$$ a finite subset $$J subset I$$ such that
$$K cap bigcap_{i in I} V_i = K cap bigcap_{i in J} V_i ,?$$