When the waist size of a Gaussian beam becomes of the order of the wavelength of light, the beam does not satisfy the paraxial condition in which it is derived. In this paper, we define the lower bound to the waist size by showing that a Gaussian beam whose waist size is larger than this bound safely satisfies the paraxial condition. A beam which is Gaussian in form but violates the paraxial condition is called a nonparaxial Gaussian beam. We clarify the range of the waist size for which the first-order correction to this beam is effective. It is shown that a distinct value of the waist size exists for which the paraxial approximation completely fails and the first-order correction never works.