On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x1,...,xl be real points from l distinct components, with l<nX. We prove that the divisor x1+......+xl is rigid. We also give a very simple proof of the Harnack's inequality