The theorem states that
$$ H(P)\leq\mathrm{MinACL}(P)<H(P)+1 $$
where, $\mathrm{MinACL}$ means the minimum average code word length of a given information source, i.e. the average code word length of any Huffman coding and $H$ means the entropy of the probability distribution $P$.
Now, the problem is how to show that for any $\epsilon>0$, there is a probability distribution $P$ s.t. $\mathrm{MinACL}(P) - H(P)\geq1-\epsilon$?
(I was given a hint that I can start with a source s.t. $H(P)=\mathrm{MinACL}(P)$ and try to change the probabilities in order to skew the code.}