In this letter, we present a closed-form bound of the average bit error rate (BER) performance for the multi-hop amplify-and-forward (AF) relaying systems with fixed gain in Rayleigh fading channel. The proposed bound is derived from the probability density function (PDF) of the overall multi-hop relay channel under the assumption of asymptotic high signal-to-noise ratio (SNR) at every intermediate relays. When intermediate relays actually operate at finite SNR, the proposed BER bound becomes looser as the SNR of the last hop increases. In order to reflect the effect of all the noise variances of relay links to the BER evaluation, a hop-indexed recursive BER approximation is proposed, in which the proposed bound of BER under asymptotic high SNR is used. The simulation results manifest that the proposed hop-indexed recursive BER approximation can not only guarantee accuracy regardless of the SNR of the last hop but also provide higher accuracy than the previous works.