The punctured euclidean space is simply connected (dim>2)

I'm looking for a proof of the statement that $\displaystyle \mathbb{R}^n\backslash \{0\}$ is simply connected.

I've already got a proof, but I'm trying to find another one (one that uses the fact that a product of simply connected space is again simply connected).

Proof: Any loop $\displaystyle \gamma$ in $\displaystyle \mathbb{R}^n\backslash \{0\}$

can be homotopied to a path on the unit sphere $\displaystyle S^{n-1}$ by $\displaystyle \gamma_t(s)=\gamma(s)/||\gamma(s)||^t$. Paths on the sphere $\displaystyle S^{n-1}$ can be homotopied to a point for $\displaystyle n>2$