Let $A$ be a random variable that takes each value $a$ in the finite alphabet $\mathcal{A}$ with probability $p(a)$. Show that, if each $l(a)$ is an integer greater than 0 and $\sum 2^{-l(a)} \leqslant 1$, then there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ with each codeword $c(a)$ having length $l(a)$.

Prove that, for any decodable code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$, we have

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

where $H(A)$ is the entropy of the random variable $A$. When is there equality in this inequality?