Counterexample to a set with a unquie maximal element implying existence of a largest element

Question

In Willard's General Topology, he defines a maximal element to be an element of a set $A$ provided $(\forall b \in A) \ \ b_1 \leq b \implies b_1=b $. He provides this afterwards:

I don't see how $b$ is a maximal element. For instance, we have $b \leq a_6$, which certainly doesn't imply $b = a_6$. Am I missing something?