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.