We utilize the integrality conjecture to show that the torus partition function of a fermionic rational conformal theory in the Ramond-Ramond sector becomes a constant when the bound ${h}^{R}\ensuremath{\ge}\frac{c}{24}$ is satisfied, where ${h}^{R}$ denotes the conformal weights of Ramond states and $c$ is the central charge. The constant-valued Ramond-Ramond partition function strongly suggests the presence of supersymmetry unless a given theory has free fermions. The lower bound ${h}^{R}\ensuremath{\ge}\frac{c}{24}$ can then be identified with the unitarity bound of $\mathcal{N}=1$ supersymmetry. We thus propose that, for rational CFTs without free fermions, $({h}^{R}\ensuremath{-}c/24)\ensuremath{\ge}0$ can imply supersymmetry.