  In this note we consider long range $q$-states Potts models on
$\mathbf{Z}^d$, $d\geq 2$. For various families of non-summable ferromagnetic
pair potentials $\phi(x)\geq 0$, we show that there exists, for all inverse
temperature $\beta>0$, an integer $N$ such that the truncated model, in which
all interactions between spins at distance larger than $N$ are suppressed, has
at least $q$ distinct infinite-volume Gibbs states. This holds, in particular,
for all potentials whose asymptotic behaviour is of the type $\phi(x)\sim
\|x\|^{-\alpha}$, $0\leq\alpha\leq d$. These results are obtained using simple
percolation arguments.
