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*By the randomness of the code construction, we can assume that the average probability of error averaged over all codes does not depend on the index sent. Thus, without loss of generality, we can assume W = 1.
*From the joint AEP, we know that the probability that no jointly typical X exists goes to 0 as n grows large. We can bound this error probability by <math>\epsilon</math>.
*Also from the joint AEP, we know the probability that a particular <math>X_1^n(i)</math> and the <math>Y_1^n</math> resulting from W = 1 are jointly typical is <math>\le 2^{-n(I(X;Y) - 3\epsilon)}</math>. Thus,
Define
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<math>E_i = \{(X_1^n(i), Y_1^n) \in A_\epsilon^{(n)}\}, i = 1, 2, ..., 2^{nR}</math>
<math>P(error) = P(error|W=1) \le P(E_1^c) + \sum_{i=2}^{2^{nR}}P(E_i)</math>
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