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MATH 0413 (29)
Pan Yibiao (29)
Lecture 25

# MATH 0413 Lecture 25: math-0413-lecture-notes-25 Premium

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School
University of Pittsburgh
Department
Mathematics
Course
MATH 0413
Professor
Pan Yibiao
Semester
Spring

Description
02/27/17 Them . let { an }nF una { b}M be Two convergent sequences , . , win Vb an E bn yon WU n E IN then win an 1 bn . n→x n→x Pth = - tn E IN . We Define {ew3nF by cn bn an By an 4 bn , , have 0 cn I . Since { E { bn are both { is an } } convergent , on } convergent ann win on = win - win . bn an n→x nax n→x = - < 0 , let E ⇒ Eo > 0 and Suppose wnwg.cn . wnino .cn win an = - Eo . v. . - f- 3g def . of Limit I k . EIN sit |Cn E° ) I L E when nieko . , . 1 Q Ch + E . I C E . when n I ko 9 - < Ee cut Eo < E . when h I ko a cn < 0 when n 2 k . This contradicts the statement that on 20 the 7N . Thus , - him one 0 whriw means that bn an ? 20 -9 . inning hnin→• n→x win him bn I an . n→x n → a tfn : general an < bn ⇒ wing an wnm→•bn . : Counterexample let an =0 , bn - it n EIN . Then > =0 < = bn th EIN . f. an In win 0 - 0 = =0 i Thus Lying In , noirs . um * Lin bn . n→x n→x However we can That an < bn ⇒ an < bn ⇒ win an E , say → n A THEIN Win
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