sols4 - 4. SEPTEMBER 8TH: MORE EXAMPLES. TECHNIQUES OF...

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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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sols4 - 4. SEPTEMBER 8TH: MORE EXAMPLES. TECHNIQUES OF...

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