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Di Bartolo, Dolgert and Dorsey have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model in the weak-$\kappa$ limit. These authors deduced a formal expansion for the superheating field $h^{sh}(\kappa)$ up to order four, extending the de Gennes formula and the two terms in Parr's formula. In this paper, we get an upper bound for $h^{sh}(kappa)$ and we deduce a rigorous proof of the Parr formula

The Parr formula for the superheating field in a semi-infinite film