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The proof runs through in almost the same way as that of channel coding theorem. Achievability follows from random coding with each symbol chosen randomly from the capacity achieving distribution for that particular channel. Typicality arguments use the definition of typical sets for non-stationary sources defined in [[Asymptotic Equipartition Property]].
The technicality of [[lim inf]] comes into play when <math>\frac{1}{n}\sum_{i=1}^n C_i</math> doesn't converge.
==References==
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