Operational Channel Capacity

The operational channel capacity is the supremum over all reliable transmission rates.

Lemma

A Binary Symmetric Channel (BSC) with error probability has nonzero capacity.

Proof

Let be such that
We claim that we can Transmit Reliably at rate
Let have Size of Largest Code
Then

Replacing with a subcode, we can assume
The Maximum Error Probability satisfies:

Let such that .
Then for large we find .
So the above probability goes to 0 by Weak Law of Large Numbers and: