The main result of the paper is a simple proof that for every closed convex set \(C\) in a separable Banach space \(X\) there is a \(C^{\infty}\)--smooth nonnegative function \(f\) defined on \(X\) such that \(f^{-1}(0)=C.\) The starting point is to write \(C\) as a countable intersection of closed half--spaces. As interesting corollaries, the authors come to easy proofs that every bounded closed set in \({\mathbb R}^n\) is a Hausdorff limit of \(C^{\infty}\)--smooth convex bodies, and, in a separable Banach space, every closed convex subset can be approximated in the sense of Mosco by \(C^{\infty}\)-smooth convex bodies.