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Definition: The supremum of f is the smallest upper bound of f:
sup(f) := such S: S is upper bound of f /\ (forall S': S' is upper bound of f => S <= S').
The infimum of f is the greatest lower bound of f:
inf(f) := such I: I is lower bound of f /\ (forall I': I' is lower bound of f => I >= I').