[微積] 高微 about lim sup Yn

[pop10353 (卡卡:目)]

Q:
    Let Xn be a bounded sequence of real numbers and
Yn= [(-1)^n]Xn.
    Show that lim sup Yn ≦ lim sup |Xn|.

<sol>

Since Yn= [(-1)^n]Xn  => Yn= ±|Xn|
                      => Yn≦|Xn|  for all n in N
                      => So Yn is bounded sequence in R.

Let b= lim sup Yn. Then there is a subseqence Yn_k converge to b.
=> For any ε>0 ,then b-ε < Yn_k ≦ |Xn_k| for the large enough k.
=> Xn is bounded ,then |Xn| is bounded.
*=> Then |Xn_k| is bouned and must have a cluster point,and
they must all as large as b-ε.
*=> Since ε is arbitrary, they must all as large as b.
*=> So lim sup Yn = b ≦ lim sup |Xn|.


想請問打*號的最後幾句 是怎麼推過去的??

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◆ From: 124.9.202.243
推 chy1010     :Yn_k 收斂至 b, |Xn_k| 是 Yn_k 取絕對值
→ chy1010     :當然會有子數列收斂 (have a cluster point)
→ chy1010     :第二個, \epsilon 可以 take limit 到 0, 保持不等號
→ chy1010     :就會有第三個 *
推 james2009   :柯西
→ pop10353    :請樓上大大 開示@@