Let a and b be ideals of a ring A. Define $$ab=\left\{{\sum_{j=1}^{n} a_jb_j|a_j\in a,b_j \in b,n \in \mathbb{N}}\right\}$$ Prove that $ab$ and $a\cap b$ are ideals of A, and that $a\cap b \supseteq ab$ but they are not necessarily equal.
For the first bit to be proved, I assume it is necessary to simply show the properties of an ideal hold for both $ab$ and $a\cap b$, but how do you show that $a\cap b \supseteq ab$?
Note that $a_jb_j\in a$ and $a_jb_j\in b$ for each $j$