This paper proves that the frequency response gains of fast-sample/fast-hold approximations of a sampled-data system converge to that of the original system as the sampling rate gets faster. While this may appear to hold trivially, there is a serious technical difficulty, and the proof is indeed nontrivial. It is also guaranteed that this convergence is uniform on the total frequency range. The latter property is necessary to guarantee that a single approximant can be used for frequency response computation for the overall frequency range.