It is often said that the achievability proof for Shannon's coding theorem relies on the channel being discrete and memoryless. At the same time, following the classical proof (using random coding and AEP), I can't find any part of the proof that explicitly uses the fact that the channel is memoryless (except arguably perhaps the guaranteed existence of a supremum of the mutual information?), as AEP has been shown to work for any stationary and ergodic channel. Further, Shannon's original paper does not seem to make any mention of memory in the proof as far as I can tell.
Is it true that the classical achievability proof relies explicitly on the channel being memoryless? If so, where in the proof is the memoryless assumption used?