I succeeded in showing that adversarial indistinguishability implies perfect secrecy by using Lemma 2.4 from the book and constructing a probabilistic adversary with advantage $\neq 1/2$.
• Let $A(r)$ denote the behavior of a probabilistic adversary $A$ when its random coins are fixed to $r$. So each $A(r)$ is a deterministic adversary. Now if $A$ is a good adversary on average, it means $\Pr_r[A(r) \mbox{ succeeds}] > \epsilon$. But then there must be some fixed $r^*$ for which $A(r^*)$ succeeds with probability at least $\epsilon$. So an adversary might as well just run deterministic strategy $A(r^*)$. – Mikero Oct 20 '16 at 17:42