Jacka Lu 發問時間: 科學數學 · 1 0 年前

國外友人請我幫問數學題目- 6

6. Let S(n) and T(n) be bounded sequences. Prove that lim inf(S(n) + T(n))≧ lim inf S(n) + lim inf T(n). Produce an example in which the inequality is strict.

1 個解答

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  • prime
    Lv 4
    1 0 年前
    最佳解答

    Let

    An = { T(i) + S(i) : i is integer and i ≧ n }

    Bn = { T(i) : i is integer and i ≧ n }

    Cn = { S(i) : i is integer and i ≧ n }

    1. claim: inf An ≧ inf Bn + inf Cn.

    For each i ≧ n, T(i)≧inf Bn and S(i)≧ inf Cn.

    Then S(i)+T(i) ≧ inf Bn + inf Cn for each i ≧ n.

    It means inf Bn + inf Cn is a lower bound of An.

    So we have inf An ≧ inf Bn + inf Cn.

    Taking limit on both sides, it means lim inf An ≧ lim inf Bn + lim inf Cn.

    2. Let S(n)=(-1)^n, T(n)=(-1)^(n+1). We have S(n)+T(n)=0 for all n.

    Moreover, lim sup S(n) = lim sup T(n) = - 1

    Then lim sup S(n)+T(n) = 0 > - 2 = lim sup S(n) + lim sup T(n)

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