Runge tubes in Stein manifolds with the density property

In this paper we give a very simple proof of the existence and plenitude of Runge tubes in ${\mathbb{C}}^n$ $(n>1)$ and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold $A$ of codimension at least two in a complex Euclidean space ${\mathbb{C}}^n$, the normal bundle of $A$ in ${\mathbb{C}}^n$ admits a holomorphic embedding onto a Runge domain in ${\mathbb{C}}^n$ which agrees with the inclusion map $A\hookrightarrow {\mathbb{C}}^n$ on the zero section.