I must prove the following:
Let $X_i \not= \emptyset, \ \forall i \in I$, and assume $\prod_i X_i$ is normal. Then
1) $X_i$ is $T_1$ for all $i$
2) $X_i$ is $T_4$ for all $i$
The initial product topology is the coarsest topology for which all canonical projections $\pi_i: X \mapsto X_i$ are continuous. $T_1$ means that for any two distinctive points, both have an open neighborhood that doesn't contain the other. Equivalent to that any singleton set is closed. $T_4$ is the same as normality and means that any two disjoint closed sets have disjoint open neighborhoods containing them.
Normality is not hereditary, so that cannot be used.
I considered using preimages. Because the projections are continuous, the preimage of any closed set should be closed. But I don't know what sets are closed in $X_i$.