Unformatted text preview: S ⊆ R be nonempty. Show that if u = sup S , then for every number n ∈ N the number u-1 n is not an upper bound of S , but the number u + 1 n is an upper bound of S . d To prove a “for every n ∈ N ” statement, prove it for a ﬁxed n (but do not specify a particular example), and then say that your proof will hold for all of them, since the only assumption you used about n was that it was in N . Since upper bound and supremum are words that we have formally deﬁned, you should use the deﬁnitions, or the theorems and lemmas that we proved using the deﬁnitions. In particular, to prove that u + 1 n is an upper bound, you must show that it satisﬁes a deﬁnition of upper bound. In other words, you must show that for every s ∈ S , we have s ≤ u + 1 n . Upshot: Assume u = sup S . Let n ∈ N . Use the deﬁnitions of supremum and upper bound to show that u-1 n is not an upper bound and u + 1 n is an upper bound....
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- Spring '08
- Supremum, Order theory, Quantification, Universal quantification, Existential quantification, upper bound