jimlucky的部落格

In Manfred Stoll
Limsup Sn is defined by use
b1,b2,....bi...bn
where bi = sup{Si,Si+1,Si+2....}
then limsup Sn =lim bn.....(*)
(this is seem a thm in Apostol)
and in Rudin
limsup Sn is defined by
E={collection all subsequential limits }
limsup Sn = supE
and liminf Sn = infE....(**)
consider (*)
(*)are equivalent to (**) clearly
----------------------------------------
Theorem
(a)Suppose limsup Sn belongs real number
then limsup Sn = x
if and only if
for any E>0
(i)there exist n0 such that Sn =n0
(ii)Given n is natrual number
there exists k belong to N with k>=n such that Sk >x-E
(b)limsup Sn is infinite
if and only if
given M and n belong to N,
there exists k belong to N with k >=n
such that Sk >= M
(C)limsup Sn is negative infinite
if and only if
Sn--> negative infinite as n-->infinite
--------------------------------------------
use(*)to prove theorem(a)
(=>)(i)
suppose B=limsup Sn = lim bk
where bk =sup{Sn|n>=K}
let E >0
since lim bk =B
there exists a positive integer n0 s.t
bk < B+E for all k>=n0
since bk >= Sn for n >= k
so Sn < B+E for all n >=n0
(ii)
Suppose n belong to N is given
since bk-->B
and {bk}is monotone decreasing
bk >=B for all k
in particular bn >=B
since bn =sup{Sn,Sn+1.....}
Given E >0
there exists an integer k >=n s.t
Sk > bn-E >=B-E
(<=)assume that (i)(ii)hold.
Let E >0 be given
By(i)there exists n0 s.t Sn < B+E for all n>=n0
therefore, bn0 =< B+E
Since bn is monotone decreasing
bn =< B+E for all n>=n0
thus limsup Sn =lim bn =< B+E
then limsup Sn =< B
Suppose B`=limsup Sn < B
choose E >0 s.t B` =< B-2E
by(i) we know exist n0 s.t
Sn < B`+E < B-E for all n >=n0
which contradicts (ii)
thus limsup Sn=B
QED