Let $L/K$ be a finite extension of number fields. Let $I$ be an ideal in $L$, we define the norm $N(I)$ of $I$ to be the ideal in $K$ generated by elements of the type $N_{L|K}(a)$ where $a \in I.$ I want to show that $N$ is multiplicative. I am able to show that $N(I)N(J) \subseteq N(IJ)$ because, elements in $N(I)$ looks like $\sum r_i N(a_i)$ where $a_i \in I,$
So if $x=\sum_i r_iN(a_i), a_i \in I$ and $y= \sum_j s_j N(b_j), b_j \in J$ then $$xy = \sum \sum r_i s_j N(a_i b_j) \in N(IJ).$$ So $$N(I)N(J) \subseteq N(IJ).$$
What is not clear to me is the other direction. Since, norm is not additive in general, it is not clear to me how to prove this.
Note : Ideals in $L$ means ideals in the ring of integers of $L.$
Localize. Prove the norm map on ideals commutes with localization at prime ideals: $$ {\rm N}_{\mathcal O_L/\mathcal O_K}(I)\mathcal O_{K,\mathfrak p} = {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(I\mathcal O_{L,\mathfrak p}) $$ for all nonzero prime ideals $\mathfrak p$ in $\mathcal O_L$.
Thus proving that $$ {\rm N}_{\mathcal O_L/\mathcal O_K}(IJ) = {\rm N}_{\mathcal O_L/\mathcal O_K}(I){\rm N}_{\mathcal O_L/\mathcal O_K}(J) $$ where $I$ and $J$ are two (nonzero) ideals in $\mathcal O_L$ is equivalent proving that $$ {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(I\mathcal O_{L,\mathfrak p}J\mathcal O_{L,\mathfrak p}) = {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(I\mathcal O_{L,\mathfrak p}){\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(J\mathcal O_{L,\mathfrak p}) $$ for all nonzero prime ideals $\mathfrak p$ in $\mathcal O_L$. At this level of generality, you should aim to prove a more general result: $$ {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(\mathfrak a\mathfrak b) = {\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(\mathfrak a){\rm N}_{\mathcal O_{L,\mathfrak p}/\mathcal O_{K,\mathfrak p}}(\mathfrak b) $$ for all nonzero ideals $\mathfrak a$ and $\mathfrak b$ in $\mathcal O_{L,\mathfrak p}$. That is, prove the norm map on ideals in $\mathcal O_{L,\mathfrak p}$ is multiplicative.
The advantage of the latter equation is that the base ring $\mathcal O_{K,\mathfrak p}$ is a DVR and $\mathcal O_{L,\mathfrak p}$ is a PID with finitely many prime ideals, so all ideals in $\mathcal O_{L,\mathfrak p}$ are principal.