How to prove that $R^n\setminus \{0\}$ is not contractible

Question

If I must prove that $R^{n}\setminus\{0\}$ is not contractible, how may I do so formally. Using the intuitive notion of contractibility at a point as being that any surface homeomorphic to an $n$ sphere, I was thinking that it could be proven by contradiction as follows: Assume for the sake of contradiction that $R^{n}\setminus\{0\}$ is contractible. Then, as $R^{n}\setminus\{0\}$ can be continuously mapped to $S^{n-1}$, $S^{n-1}$ must also be contractible. However, as the only shape homeomorphic to $S^{n-1}$ that passes through the point under consideration is $S^{n-1}$ itself, there exists no contraction at that point of an $S^{n-1}$ sphere to a point. Thus, we have reached a contradiction, and $R^{n}\setminus\{0\}$ is not contractible. If indeed the formal definition of contractibility (in terms of null homotopy)is equivalent to the one I have used, could you please tell me a source for the same? (so that I may cite it for an assignment)