Product of ideals is a subset of an intersection of ideals

Take some element $a \in IJ$, which means we have that $a$ is of the form $a = i_1j_1 + \dots + i_nj_n$. Now $i_k \in I$ for all $1 \leq k \leq n$, such that $i_kj_k \in I$ and therefore $a = i_1j_1 + \dots i_nj_n \in I$. Analogously we get $a \in J$. Thus $IJ \subset I \cap J$.