  Consider a linear program of the form $\max\;c^{\top}x:Ax\leq b$, where $A$
is an $m\times n$ integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardos
proved that, given an optimal solution $x^{*}$, if an optimal integral solution
$z^{*}$ exists, then it may be chosen such that $\left\Vert
x^{*}-z^{*}\right\Vert _{\infty}<n\Delta$, where $\Delta$ is the largest
magnitude of any subdeterminant of $A$. Since then an open question has been to
improve this bound, assuming that $b$ is integral valued too. In this
manuscript we show that $n\Delta$ can be replaced with $\frac{n}{2}\cdot\Delta$
whenever $n\geq2$. We also show that, in certain circumstances, the factor $n$
can be removed entirely.
