Unformatted text preview: 3. Let S ⊆ R be nonempty and u ∈ R . Prove that u = sup S if and only if for every n ∈ N , u + 1 /n is an upper bound of S while u-1 /n is not. Hint : For the right-to-left direction, use one of the corollaries of the Archimedean Prop-erty. Also, for the same direction, in showing that u is an upper bound, you may ﬁnd Exercise 2.3–8 helpful, while in showing that u is the least upper bound, you may ﬁnd Lemma 2.3.4 helpful....
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- Spring '08
- Sets, Supremum, Archimedean Property