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?