Canonical Injection into Cartesian Product of Modules

Theorem
Let $G$ be the cartesian product of a sequence $\left \langle {G_n} \right \rangle$ of $R$-modules.

Then for each $j \in \left[{1 \,.\,.\, n}\right]$, the canonical injection $\operatorname{in}_j$ from $G_j$ into $G$ is a monomorphism.