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Unformatted text preview: 4. SEPTEMBER 8TH: MORE EXAMPLES. TECHNIQUES OF PROOF. 9 4. September 8th: More examples. Techniques of proof. 4.1 . Are ∀ x > ∃ N : n > N = ⇒  s n  < x and ∀ x > ∃ N : n > N = ⇒  s n  < 3 x equivalent statements? If so, prove that they are. If not, find an example of an assignment s n which satisfies one of the statements but not the other. Answer: These statements are equivalent: an indexed set { s n } n ∈ N satisfies one if and only if it satisfies the other. To see this, it may be helpful to rename the variables x and N by y and M in one of the statements (of course changing the name of a quantified variable does not change the meaning of the statement 12 ). So I will compare ( † ) ∀ x > ∃ N : n > N = ⇒  s n  < x with ( ‡ ) ∀ y > ∃ M : n > M = ⇒  s n  < 3 y and prove that these two statements are equivalent. Proof. Assume { s n } n ∈ N satisfies ( † ); I will prove it satisfies ( ‡ ). To prove it satisfies ( ‡ ), let y be an arbitrary given number, and let x = 3 y . Since { s n } n ∈ N satisfies ( † ), there exists an N such that n > N = ⇒  s n  < x . Since x = 3 y , this is saying that n > N = ⇒  s n  < 3 y . Therefore, choosing M = N shows that ∃ M : n > M = ⇒  s n  < 3 y , verifying ( ‡ )....
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.
 Fall '11
 Aluffi

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