Expressing a point in $\mathbb{R}^n$ as a sum of unit vectors

Question

I'm pretty sure that any point in $\mathbb{R}^n$ can be written as a sum of finitely many unit vectors (in $\mathbb{R}^n$, of course). However, I have no idea how to go about proving this.

Any ideas?

Answer 1

Assume without loss of generality that your point $P$ is on the $x$-axis in the $xy$-plane. Let $L$ be perpendicular bisector of $OP$ where $O$ is the origin. Let $C$ be the circle centered at the origin of radius $\lceil P \rceil$. Then $L\cap C=\{A,B\}$ and $P=A+B$. Note that $A,B$ are of integer length and hence are sums of unit vectors (of the same direction). This decomposes $P$ into the sum of $2\lceil P\rceil$ unit vectors.