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 Basic Results about Modules, $$\forall \lambda: \lambda \circ e = e$$.

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 $$\sum_{k=1}^n \lambda_k \circ a_k = e$$, namely, the $$\lambda$$ which is multiplied by $$e$$.

The result follows.