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# 高微小考試題7

Let A be a non-empty subset of R and let α∈R be an upper bound of A. Prove thatα=supA if and only if for any ε＞0 there exists x∈A such thatα－ε＜x.

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- 老怪物Lv 710 years agoFavorite Answer
設 α=supA , ε＞0.

若不存在 x in A s.t. α-ε＜x, 表示 x in A implies x≦α-ε<α,

與 α=supA 矛盾.

反之, 設α為 A 之一 upper bound, 且

for all ε>0, 存在 x in A s.t. α-ε＜x.

則 for any β<α, 取 ε=α-β, 存在 x in A s.t. β<x, 故 β 不是 A 的 upper bound.

換言之, α 是 A 的所有 upper boubds 中最小的. 即: α=supA.

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