Restriction of stable rank two vector bundles in arbitrary characteristic

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) center dot Hdim(X)-2 and H-dim(X) such that the restriction of E to a general divisor in vertical bar mH vertical bar 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.