Abelian implies universal power map is endomorphism

Statement

Let $(G,+)$ be an Abelian group, and $n$ be an integer. The map $g\mapsto ng$ (i.e., the map $g\mapsto g+g+\dots +g$ done $n$ times when $n$ is positive and $(-g)+(-g)+\dots +(-g)$ done $(-n)$ times when $n$ is negative) is an endomorphism of $G$.

Proof

Given: An Abelian group $G$, an integer $n$.

To prove: The map $g\mapsto ng$ is an endomorphism of $G$: in other words, $n(g+h)=ng+nh$.

Proof: The proof basically follows from commutativity. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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