Outline of Proof
As with several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result. These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds.
The following outlines are only one set of many different styles available for study in information theory texts.
Read more about this topic: Noisy-channel Coding Theorem
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