Essential bit content - prove that we can't use less bits than that
The function wouldn't be one-to-one otherwise.
The size of the set in the left is $|A|$, and its image would be of size at most $2^l&lt;2^{H_0}=|A|$, therefore, the function wouldn't be injective and thus you wont be able to de-compress some of the data.
Therefore it will be a lossy-compression for any $l&lt;H_0$.