Let S be a set well-ordered by ≤ . Let x ∈ S. Suppose x is not maximal in S; i.e., there exists an element z ∈ S with z

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Let S be a set well-ordered by ≤ . Let x ∈ S. Suppose x is not
maximal in S; i.e., there exists an element z ∈ S with z > x.
Prove that there then exists an element s(x) in S, called the
immediate successor, of x such that i) x < s(x) and ii) if x
< z then s(x) ≤ z, as well. Hint: Consider the complement of the
set s(x) ∪ {x} in S.