Shannon's source coding theorem for a source with at least two symbols

Let $p$ be a probability distribution on a finite set $\iota$ of source symbols with at least two symbols, $p_i>0$ and $\sum_i p_i=1$, and let $\alpha$ be a code alphabet with $D=|\alpha|\ge2$ letters. Write $H_D(p)=-\sum_i p_i\log_D p_i$ for the base-$D$ entropy (the platform definition SourceCoding.entropy). Then:

1. every injective assignment of codewords $c:\iota\to\alpha^{*}$ whose codeword set is uniquely decodable has expected length at least the entropy,

2. there is an injective assignment of codewords with uniquely decodable codeword set whose expected length is strictly less than the entropy plus one,

This is Shannon's source coding theorem in the form $H_D\le L^{*}<H_D+1$ (Cover and Thomas, Elements of Information Theory, Theorem 5.4.1). The lower bound follows from the Kraft–McMillan inequality and Gibbs' inequality; the upper bound is attained by the Shannon–Fano lengths $\ell_i=\lceil\log_D(1/p_i)\rceil$, which satisfy Kraft's inequality and are positive precisely because every $p_i<1$, which is where the hypothesis $|\iota|\ge2$ is used. Without that hypothesis the second part fails for the one-symbol source (entropy $0$, but no uniquely decodable code has a codeword of length $<1$), which is why the earlier platform statement SourceCoding.shannon_source_coding_theorem was disproved.