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We present a new derivation of the source coding theorem for discrete-time sources. This proof parallels Gallager's  derivation of the random coding bound for channel coding theory and shows that the classical random coding exponent also emerges as a critical quantity for source coding. The major advantage of this approach is the simplicity of the derivation and its close relationship to the more familiar channel coding theory. The source coding theorem we derive here also yields a natural bound on the rate of convergence to the rate-distortion limit.