This paper has interest in the effect of a fine frequency offset, defined in ITU-T G.225, to the training performance of an adaptive equalizer. This paper uses Hilbert filter in configuring a transmission system model in order to let it get a frequency offset. Also additive white Gaussian noise and band-limited filter are considered. The signal received from the above transmission system applies to an adaptive equalizer with LMS algorithm, and its training procedures are investigated. As a result, we could find that even small fine frequency offset can severely deteriorate training performance of adaptive algorithm.