I have proved that if $P_1$ and $P_2$ are prime ideals and an ideal I belongs to $P_1 \bigcup P_2$. Then $I\subseteq P_1$ or $I\subseteq P_2$.
But I am not able to think how can I generalize it. ( I don't want to use induction so I didn't tried that). So, the problem is following: Let I be any ideal in a ring A such that $I\subseteq I_1 \bigcup I_2 \bigcup ... \bigcup I_n$, where each $I_i$ is prime ideal then show that $I \subseteq I_m$ for some $m\in$ {$1,...,n $}.
Please shed some light on how should I approach this.