  Posterior matching (PM) is a sequential horizon-free feedback communication
scheme introduced by the authors, who also provided a rather involved
optimality proof showing it achieves capacity for a large class of memoryless
channels. Naghshvar et al considered a non-sequential variation of PM with a
fixed number of messages and a random decision-time, and gave a simpler proof
establishing its optimality via a novel Shannon-Jensen divergence argument.
Another simpler optimality proof was given by Li and El Gamal, who considered a
fixed-rate fixed block-length variation of PM with an additional randomization.
Both these works also provided error exponent bounds. However, their simpler
achievability proofs apply only to discrete memoryless channels, and are
restricted to a non-sequential setup with a fixed number of messages. In this
paper, we provide a short and transparent proof for the optimality of the fully
sequential horizon-free PM scheme over general memoryless channels. Borrowing
the key randomization idea of Li and El Gamal, our proof is based on analyzing
the random walk behavior of the shrinking posterior intervals induced by a
reversed iterated function system (RIFS) decoder.
