In class, we worked toward defining the set of real numbers by investigating suprema. Vaguely speaking, we want to fill the “gaps” in the rational numbers. Simply adding in solutions to equations like is not sufficient (this yields the algebraic numbers, which still have “gaps”). Instead, we rely on the assumptions that (1) the rationals are a subset of the reals, and (2) the Axiom of Completeness holds [see definitions section for rigorous definitions of AoC and “bounded above”]. We investigated the two facets of the definition of a supremum, which must be shown to prove that some s is the supremum of a given set A: (1) s is an upper bound of A, and (2) of all upper bounds of A, s is the least upper bound. We walked through two proofs involving suprema, relying on epsilons in the latter proof.