  Let $X$ be a smooth variety defined over an algebraically closed field of
arbitrary characteristic and $\O_X(H)$ be a very ample line bundle on $X$. We
show that for a semistable $X$-bundle $E$ of rank two, there exists an integer
$m$ depending only on $\Delta(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the
restriction of $E$ to a general divisor in $|mH|$ is again semistable. As
corollaries we obtain boundedness results, and weak versions of Bogomolov's
theorem and Kodaira's vanishing theorem for surfaces in arbitrary
characteristic.
