Optimal noiseless coding of random variables (Corresp.)

For a discrete
n
-valued random variable
X
, Leung-Yan-Cheung and Cover recently showed that the minimal expected length of one-to-one (not necessarily uniquely decodable) codes satisfies
H(X)+ 1 >L_{l:l} \geq H(X) - \log( \sum^{n}_{i-1}2/(i+2))
. A simple and direct proof of their lower bound is given which does not use the method of Lagrange multipliers.