Subset of Module Containing Identity is Linearly Dependent

Theorem
Let $G$ be a group whose identity is $e$.

Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$.

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.

Let $H \subseteq G$ such that $e \in H$.

Then $H$ is a linearly dependent set.

Proof
From Scalar Product with Identity, $\forall \lambda: \lambda \circ e = e$.

Let $H \subseteq G$ such that $e \in H$.

Consider any sequence $\left \langle {a_n} \right \rangle$ in $H$ which includes $e$.

Then there exists a $\left \langle {\lambda_n} \right \rangle$ in which not all $\lambda_k = 0_R$ such that $\displaystyle \sum_{k=1}^n \lambda_k \circ a_k = e$, namely, the $\lambda$ which is multiplied by $e$.

The result follows.