Bounds of the probability of weakly typical set

Question

Theorem 3.1.2 of the book Elements of Information theory states: \begin{align} Pr\{A_\epsilon^{(n)}\} > 1 - \epsilon \end{align} The bound becomes weaker as $\epsilon$ increases. From the definition of $A_\epsilon^{(n)}$ it seems that more and more sequences are included as $\epsilon$ increases so the probability should increase. So the two results seem to contradict each other. Obviously, the theorem will still hold but it seems to me that this is a side effect of simplifying the notation by choosing $\delta = \epsilon$ in the proof below. Am I correct in saying that in reality the probability increases as $\epsilon$ increases, or am I misunderstanding something?