Cici Lin 發問時間: 科學數學 · 1 0 年前

請問厲害的大大”高微”問題

Let {Xn} be sequence in R.

(a) Xn is bounded below, liminf{Xn}= a iff (i) for allε>0, exist N s.t. a-ε<Xn, n≧N. (ii) for all ε>0, for all M, exist n≧M s.t. Xn<a+ε.

(b) Xn is bounded above, limsup{Xn}= b iff (i) for all ε>0, exist N s.t. Xn<b+ε, n≧N. (ii) for all ε>0, for all M, exist n≧M s.t. b-ε<Xn.

請幫証"" <= ""謝謝

2 個解答

評分
  • prime
    Lv 4
    1 0 年前
    最佳解答

    Let An={Xn,Xn+1,....}.

    (a)

    For a given ε>0, there exist N1 s.t. a-ε<Xn for all n≧N1. ( by (1))

    We have a-ε is a lower bound of A_k for all k≧N1.

    It implies a-ε ≦ inf A_k for all k≧N1. ----- (*)

    For each k≧N1, there exists an Nk≧ j s.t. X_(Nk)<a+ε.(by (2))

    Then inf Ak ≦ X_(Nk) <a+ε for all k≧N1. ----(**)

    Combiming (*) and (**), a-ε≦ inf A_k < a+ε for all k≧N1.

    Thus, | inf A_k - a | ≦ ε for all k≧N1 .ie. liminf Xn = a.

    (b)

    For a given ε>0, there exist N1 s.t. b+ε>Xn for all n≧N1. ( by (1))

    We have b+ε is a upper bound of A_k for all k≧N1.

    It implies b+ε≧ inf A_k for all k≧N1. ----- (*)

    For each k≧N1, there exists an Nk≧ j s.t. X_(Nk)>b-ε.(by (2))

    Then b-ε< X_(Nk)≦ sup Ak for all k≧N1. ----(**)

    Combiming (*) and (**), b-ε< sup A_k ≦ b+ε for all k≧N1.

    Thus, | sup A_k - b | ≦ ε for all k≧N1 .ie. limsup Xn = b.

    2006-11-26 01:35:22 補充:

    (b)It implies b+ε≧ sup A_k for all k≧N1. ----- (*)

    2006-11-26 20:18:12 補充:

    lim sup的定義不是令 Rn = sup{X_n, X_(n+1), X_(n+2)............}limsup Xn = lim Rn我這邊是假設 Let An={Xn,Xn+1,....}.則 Rn = sup An直接去處理集合An會比較好證明....

    2006-11-26 20:20:43 補充:

    對不起打錯了, 那應該是k.

    2006-11-26 20:23:09 補充:

    For each k≧N1, there exists an Nk≧ k s.t. X_(Nk)<a+ε.(by (2))Then inf Ak ≦ X_(Nk) <a+ε for all k≧N1. ----(**)Combiming (*) and (**), a-ε≦ inf A_k < a+ε for all k≧N1.Thus, | inf A_k - a | ≦ ε for all k≧N1 .ie. liminf Xn = a.

    2006-11-27 01:53:10 補充:

    不是同一個 基本上這兩個證明幾乎完全一樣!你只要把對應數字的不等號反過來,上界改下界,a改成b.證明就轉過去了.其實如果{Xn}滿足(a), 則{-Xn}滿足(b).

  • 1 0 年前

    請問A_k是sequence Xn的等k項嗎

    2006-11-26 20:09:01 補充:

    請問~~~你解答裡的""j""是什麼?

    2006-11-27 00:45:39 補充:

    請問(a)中的N1與(b)中的N1是一樣的嗎

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