u = sup(S) iff for all v ∈ R, if v ﹤ u then there is some s ∈ S such that v ﹤ s, & u is upper bound

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u = sup(S) iff for all ε ﹥ 0, there exists s ∈ S such that u - ε ﹤ s, & u is an upper bound (Proof)

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