Re: [問題] 收斂問題
Ｄ Pr
Suppose Ｘ_n ──→ Ｘ and Ｙ_n ──→ ０, then
Ｄ툊 Ｘ_n＋Ｙ_n ──→ Ｘ
[pf]
Let x be a point of continuity of Ｆ (x).
Ｘ
Let ε＞0 be given where ε is sufficiently small. We have
┌ ┐ ┌ ┐
P│Ｘ_n＋Ｙ_n≦x│＝P│{Ｘ_n＋Ｙ_n≦x}∩{x－Ｘ_n≧－ε}│＋
└ ┘ └ ┘
┌ ┐
P│{Ｘ_n＋Ｙ_n≦x}∩{x－Ｘ_n＜－ε}│
└ ┘
┌ ┐ ┌ ┐
≦P│Ｘ_n≦x＋ε│＋P│Ｙ_n＜－ε│
└ ┘ └ ┘
┌ ┐
≦Ｆ (x＋ε)＋P││Ｙ_n│＞ε│
Ｘ_n └ ┘
we see that
___ ┌ ┐
lim P│Ｘ_n＋Ｙ_n≦x│≦Ｆ (x＋ε)
n→∞ └ ┘ Ｘ
Also,
┌ ┐ ┌ ┐
P│Ｘ_n＋Ｙ_n＞x│＝P│{Ｘ_n＋Ｙ_n＞x}∩{x－Ｘ_n≧ε}│＋
└ ┘ └ ┘
┌ ┐
P│{Ｘ_n＋Ｙ_n＞x}∩{x－Ｘ_n＜ε}│
└ ┘
┌ ┐ ┌ ┐
≦P│Ｙ_n＞ε│＋P│Ｘ_n＞x－ε│
└ ┘ └ ┘
┌ ┐
≦P││Ｙ_n│＞ε│＋1－Ｆ (x－ε)
└ ┘ Ｘ_n
┌ ┐ ┌ ┐
=> P│Ｘ_n＋Ｙ_n≦x│≧Ｆ (x－ε)－P││Ｙ_n│＞ε│
└ ┘ Ｘ_n └ ┘
┌ ┐
=> Ｆ (x－ε)≦ lim P│Ｘ_n＋Ｙ_n≦x│
Ｘꈠ ￣￣ └ ┘
Thus,
┌ ┐
Ｆ (x－ε)≦ lim P│Ｘ_n＋Ｙ_n≦x│
Ｘ ￣￣ └ ┘
___ ┌ ┐
≦ lim P│Ｘ_n＋Ｙ_n≦x│≦Ｆ (x＋ε)
└ ┘ Ｘ
Letting ε↓0 gives us the desired result.
這樣寫 ok 嗎 ^^謝謝
※ 編輯: b218h 來自: 220.139.147.70 (01/20 06:30)
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