Finite_Submodule_of_Function_Space

Let $\left({G, +}\right)$ be a group whose identity is $e$. Let $R$ be a ring. Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $S$ be a set. Let $G^S$ the set of all mappings $f: S \to G$. Let $G^{\left({S}\right)}$ be the set of all mappings $f: S \to G$ such that $f \left({x}\right) = e$ for all but finitely many elements $x$ of $S$. Then $\left({G^{\left({S}\right)}, +', \circ}\right)_R$ is a submodule of $\left({G^S, +, \circ}\right)_R$ where $+'$ is the operation induced on $G^{\left({S}\right)}$ by $+$.