$R^{(\infty)}$ is diffeomorphic to $R^{(\infty)}\setminus\{0\}$

We construct a one-to-one map of the topological direct sum $R^{(\infty)}$ of countably many copies of the real line onto $R^{(\infty)}\setminus\{0\}$ which is infinitely differentiable along with its inverse, in the sense of Michael–Bastiani.