  We establish a Liouville type theorem for the fractional Lane-Emden system:
  \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha
u=v^q&{\rm in}\,\,\R^N,\\ (-\Delta)^\alpha v=u^p&{\rm in}\,\,\R^N,
  \end{array}
  \right.
  \end{eqnarray*}
  where $ \alpha\in(0,1) $, $ N>2\alpha $ and $ p,q $ are positive real numbers
and in an appropriate new range.
  To prove our result we will use the local realization of fractional
Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a
degenerate elliptic equation in the spirit of Caffarelli and Silvestre
\cite{CS}. Our proof is based on a monotonicity argument for suitable
transformed
  functions and the method of moving planes in an infinity half cylinder based
on some maximum principles which obtained by some barrier functions and a
coupling argument using fractional Sobolev trace inequality.
