When Huffman coding is inefficient? I have a question regarding the redundancy of Huffman coding. I know that for a general prefix code we have the following inequality: 

$$ H(X) \le R \le H(X) + 1 $$ 

$R$ being the rate (average codeword length) and $H$ is the entropy. Based on this relation, how can we conclude that Huffman coding is very inefficient if entropy of the source is much smaller than 1 bit/symbol?
If $X$ is any non-constant source, then any codeword in any prefix code for $X$ has length at least $1$, and so $R \geq 1$. Therefore if $X$ is a source with very low entropy then there's a large discrepancy between $H(X) \approx 0$ and $R \approx 1$.

As a concrete example, consider a binary source in which the probability of one of the options is $\epsilon$. Then $H(X) = \epsilon \log (1/\epsilon) + O(\epsilon)$ whereas $R = 1$.