# algebraic topology – Free abelian groups question lemma

Definition:
An abelian group $$G$$ is free with basis $$B$$ if $$G=bigoplus_{bin B}langle brangle$$ and each $$langle b rangle$$ is infinite cylic for each $$bin B$$.

Lemma:
If $$G$$ is free with basis $$B$$ then for each $$xin G$$, $$x$$ has a unique expression $$sum m_bb$$ where $$m_bin mathbb{Z}$$ and $$m_b=0$$ for all but finitely many $$bin B$$

My attempt: Let $$gin G$$, as $$G=bigoplus_{bin B}langle brangle$$, $$g=(g_b)_{bin B}$$, where $$g_b=0$$ for all but finitely many $$b$$ and each $$g_b=m_bb$$ for $$m_bin mathbb{Z}$$.

Hence $$g=sum_{bin B}(0,…….,m_bb,0……,0)$$

However, I am unable to show that $$(0……..,m_bb,0……,0)=m_bb$$.

How does one show this?