  Let $A \in \mathbb{R}^{n \times n}$ be invertible, $x \in \mathbb{R}^n$
unknown and $b =Ax $ given. We are interested in approximate solutions: vectors
$y \in \mathbb{R}^n$ such that $\|Ay - b\|$ is small. We prove that for all $0<
\varepsilon <1 $ there is a composition of $k$ orthogonal projections onto the
$n$ hyperplanes generated by the rows of $A$, where $$k \leq 2
\log\left(\frac{1}{\varepsilon} \right) \frac{ n}{ \varepsilon^{2}}$$ which
maps the origin to a vector $y\in \mathbb{R}^n$ satisfying $\| A y - Ax\| \leq
\varepsilon \cdot \|A\| \cdot \| x\|$. We note that this upper bound on $k$ is
independent of the matrix $A$. This procedure is stable in the sense that
$\|y\| \leq 2\|x\|$. The existence proof is based on a probabilistically
refined analysis of the Random Kaczmarz method which seems to achieve this rate
when solving for $A x = b$ with high likelihood.
