Let $I$ a closed ideal of $C_0(X)$. Show that if $f\in I$, then $f^*=\overline{f}\in I$.
Tip: Let a sequence $(f_n)_{n=1}^\infty$ by $f_n=(f^*f)^{1/n}$, show that $f_n\in I$ and $\lim_{n}f_nf^*=f^*$.
I showed that $f_nf^*\longrightarrow f^*$. But the part to show that $f_n\in I$ I'm stuck. First I proved that $f^*f\in I$, 'cause $I$ is ideal, but how about $(f^*f)^{1/n}$?