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Lecture 25
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University of Pittsburgh
Mathematics
MATH 0413
Pan Yibiao
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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