Chapter 2 Sequences in R
2.1. Limits of sequences
Def: A sequence (infinite sequence) is a function from N into R, xn=f(n), we
denote it by {xn} or {xn}n?N n?N.
2.1. Definition: Let {xn} be a sequence of R, {xn} is said to convergent to a?R if for every ?>0, ? N?N s.t We denote it by limxn=a
n??xn-aN.
1Example. Show that {} is convergent
n Proof: We show limn??Given ??0.
1=0. n1-0N. nTo find an N?N s.t
By Archimedean principle ? N?N s.t N?>1.
???1 N? for n>N, we have 11 0
To find N?N s.t xn?1?1?1??. Take N=1 Then
xn-1=0N.
?limxn=1
n??Exercise. Show that {
1} is convergent 2n
2.3 Ex:ample
Show {(-1)n} has no limit. Proof: suppose lim(-1)n=a.
n??Given ?=
1 ?N?N s.t 21(-1)n-a< for n>N.
2Let n>N be even, then n+1 is odd. 2=1n-(-1)n?1
=(1n?a)?[(?1)n?1?a]?(?1)n?a?(?1)n?1?a<
11+=1 22?{(-1)n} has no limit
Theorem: Let a,b?R. If a?b?? for any ?>0 ?a=b. Proof: Suppose a?b.
Let L=a?b>0 and let ?= L=a?b
L ?? 2L , by assumption, we have 2 ?a=b 2.4:. Remark
A sequence can have at most one limit.
Proof: Let {an} be a sequence of R and let liman=L1 and liman=L2
n??n?? It is sufficient to show L1= L2. Let?>0.
?liman=L1 and liman=L2
n??n?? ?? N1?N s.t an?L1??2 for n> N1
similarly, ?N2?N s.t an?L2? For n>max(N1, N2) We have an?L1?for n>max(N1, N2)
?2 for n> N2
?2 and an?L2??2
L1?L2?L1?an?an?L2?L1?an?L2?an??2??2??
Since ? is arbitrary, we derive that L1= L2.
2.5, Definition.
By a subsequence of a sequence {xn}, we shall mean a sequence {xnk} where n1 2.6.Theorem Let {xn} be a sequence and {xn} converge to L. If {xnk}is a subsequence of {xn} then limxnk=L. nk??Proof:. limxnk= L nk?? Given ??0, we must find a k?N s.t xnk?L?? for nk>k. ?limxn?L n?? ??N?N s.t xn?L?? for n>N. Let k?N ? n1< n2< n3< n4… ? nk>k ? xnk?L?? for nk>k. ? limxnk=L nk??2.7 . Definition. Let {xn} be a sequence of R, {xn} is called bounded if ? M>0 s.t,xn?M for each n. 2.8. Theorem Every convergent sequence of R is bounded Proof: Let {xn} converge to L To show {xn} is bounded. We must find a M s.t xn?M
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