Let $A$ be a random variable that takes values in the finite alphabet $\mathcal{A}$. Prove that there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ that satisfies

$$H(A) \leqslant \mathbb{E}(l(A)) \leqslant H(A)+1$$

where $l(a)$ is the length of the code word $c(a)$ and $H(A)$ is the entropy of $A$.

Is it always possible to find such a code with $\mathbb{E}(l(A))=H(A) ?$ Justify your answer.