  Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over $\Z$ and
$x_m, m \in \mathbb Z$ be rational functions defined by
$x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is
even. In this short note, we prove that for any $m,k \in \Z$, $x_k$ can be
expressed as a substraction-free Laurent polynomial in $\Z[x_m^{\pm
1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for
coefficient-free rank two cluster algebras.
