Fundamental limits are achievable with countable alphabet

This paper investigates the capacity of a channel and the rate-distortion function of a source, where the alphabets of a channel and a source are uncountable/continuous. It is shown that, when the fundamental limits of the capacity and the rate-distortion function are finite, we can approach these limits with a countable signaling/output alphabet. Furthermore, these limits are also achievable by using codes with a sufficiently large finite signaling/output alphabet. This fact is important in relation to designing implementable codes approaching fundamental limits.