Words have bounded width in $SL(n,\mathbb{Z})$

We prove two results about width of words in $S{L}_n(\mathbb{Z})$. The first is that, for every $n\ge 3$, there is a constant $C(n)$ such that the width of any word in $S{L}_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $S{L}_n(\mathbb{Z})$ is at most 87.