In this book http://ukcatalogue.oup.com/product/9780199202492.do#.UYDnvZNk1bA (Liu's Algebraic Geometry book), we can find the next proposition;
Proposition 3.2.20. Let $X$ be a geometrically reduced algebraic variety over a field $k$. Let $k^s$ be the separable closure of $k$. Then $X(k^s)$ is not empty.
,and the proof of this Proposition starts with assuming $k=k^s$, $X$ is affine and integral by replacing $X$ by an irreducible open affine subset. But I cannot understand how we can make second assumption.(Throughout the book an algebraic variety is defined by a of finite type $k$-scheme)