This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ( )....
View Full
Document
 Fall '11
 Aluffi

Click to edit the document details