Definition (Achievable)

A rate $R$ is called achievable for a discrete channel if there exists a sequence of $(n,M_{n})$ fixed-length codes $\mathcal{C}_{n}$ for the channel with $$\liminf_{n\to\infty} \frac{1}{n}\log_{2}M_{n}\ge R$$and $$\lim_{n\to\infty}P_{e}(\mathcal{C}_{n})=0$$

Definition (Achievable (continuous))

A rate is called achievable for the Gaussian channel with power constraint $P$ if there exists a sequence of $(n,M_{n})$ block codes $\mathcal{C}_{n}$ for the channel whose codewords satisfy the power constraint such that $$\liminf_{n\to\infty} \frac{1}{m}\log_{2}M_{n}\ge R$$ and $$\lim_{n\to\infty}P_{e}(\mathcal{C}_{n})=0$$or that as the block length approaches infinity our rate approaches the optimal rate and the Code Reliability (Continuous) goes to zero.