Prove that $\mathbb{R}^n\setminus \{0\} $ is connected for $n > 1$ 
Prove that $\mathbb{R}^n\setminus \{0\} $ is connected for $n > 1$.

I don't understand where to start proving this since
$$\mathbb{R}^n \setminus \{0\} = (-\infty,0)^n \cup (0, \infty)^n $$
Which is the union of two disjoint nonempty open sets, so it can't be connected. Obviously I won't be told to prove something is true that isn't so I know I'm missing something.
We have proven that $\mathbb{R}^n$ is connected using this theorem:
Let S be a topological space, and let $T_0$ and $\{T_w\}_{w\in W} $ be connected subsets of S. Assume that $T_0 \cap T_w \neq \emptyset $ for each w. Then $T_0 \cup \left( \cup_{w \in W} T_w \right)$ is connected.
Using the first connected set {0} and the indexed ones as lines that go through the point {0} indexed by the unit sphere. 
I was hoping to do something similar with this problem, but I can't see a way to do that. Help would be appreciated. And apologies for any Latex mistakes. I'll try to fix them but I'm on vacation and only have my phone currently.

$n$-dimensional polar coordinates give rise to a continuous surjection from the cartesian product of connected intervals to $\mathbb R^n \setminus \{0\}$.
