A simple proof that time-invariant convolutional codes attain capacity

It is well known that time-varying convolutional codes can achieve the capacity of a discrete memoryless channel. The time varying assumption is needed in the proof to assure pairwise independency between the codewords. We provide a relatively simple proof that indeed time-invariant convolutional codes can achieve the capacity without any restriction (albeit, the error exponent achieved by our proof may not be the optimal)