Re: [分析] 極限/數列/機率
※ 引述《GSXSP (Gloria)》之銘言：
: Let
: M n n
: a_n = Σ | Π x_{ik} - Π y_{ik} |
: ik=1 k=1 k=1
: k=1,..n
: where
: M M
: Σ x_i = Σ y_i = 1 , x_i, y_i >= 0.
: i=1 i=1
: For instance,
: M
: a_1 = Σ | x_{i1} - y_{i1} | ,
: i1=1
: a_2 = Σ Σ | x_{i1} x_{i2} - y_{i1} y_{i2} | .
: i1 i2
: Prove or disprove that
: For all ε>0, there exist δ>0
: such that sup_n a_n < ε if a_1 < δ.
假設已經找到δ<1
let x_1 = 1, y_1 = 1-δ/3, y_2 = δ/3, 其他x, y皆為0
滿足 Σ x_i = Σ y_i = 1, x_i, y_i非負且a_1 < δ
則a_n >= |x_1^n - y_1^n| = 1 - (1-δ/3)^n 趨近於1
表示ε<1 時不可能有sup_n a_n < ε
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