Let $x\in \Bbb{R^n}$. We may claim that there is an open $U$ containing $x$. For the proof, is it enough to define a diffeomorphism $h:U\to\emptyset$? This would mean that $$h(U\cap \{x\})=V\cap(\Bbb{R^0}\times\ \{0\})$$
and if we let $V=\emptyset$ also, then $$h(U\cap \{x\})=\emptyset$$
EDIT : Okay, I realize now, that generally proving things are manifolds by the definition is probably not practical. That being said, I would be interested to know whether a "proof by definition" is feasible. Intuitively this seems very simple. Every point of $\Bbb{R^n}$ resembles the origin, which is just a point.