  Inspired by Fr\"{o}hlich-Spencer and subsequent authors who introduced the
notion of contour for long-range systems, we provide a definition of contour
and a direct proof for the phase transition for ferromagnetic long-range Ising
models on $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on a
multi-scale analysis, works for the sharp region $\alpha>d$ and improves
previous results obtained by Park for $\alpha>3d+1$, and by Ginibre, Grossmann,
and Ruelle for $\alpha> d+1$, where $\alpha$ is the power of the coupling
constant. The key idea is to avoid a large number of small contours. As an
application, we prove the persistence of the phase transition when we add a
polynomial decaying magnetic field with power $\delta>0$ as $h^*|x|^{-\delta}$,
where $h^* >0$. For $d<\alpha<d+1$, the phase transition occurs when
$\delta>d-\alpha$, and when $h^*$ is small enough over the critical line
$\delta=d-\alpha$. For $\alpha \geq d+1$, $\delta>1$ it is enough to prove the
phase transition, and for $\delta=1$ we have to ask $h^*$ small. The natural
conjecture is that this region is also sharp for the phase transition problem
when we have a decaying field.
