[問題] 一題機率論的定理證明

[buttermilk (脫脂牛奶)]

Let {X_n} be a sequence of random variables bounded in probability and

let {Y_n} be a sequence of random variables which converge to 0

                                P
in probability. Then  X_nY_n ------> 0.

Proof:

       Letε> 0 be given. Choose B_ε> 0 and an integer N_ε such that

n≧N_ε => P[∣X_n∣≦B_ε]≧1-ε.

       Then,
 ---                       ---
 lim P[｜X_nY_n｜≧ε]≦   lim P[｜X_nY_n｜≧ε,∣X_n∣≦B_ε]
 n→∞                     n→∞
                           ---
                         + lim P[｜X_nY_n｜≧ε,∣X_n∣＞B_ε]
                           n→∞
                           ---
                      ＜   lim P[｜Y_n｜≧ε/B_ε] + ε = ε
                           n→∞

From which the desired result follows.

------------------------------------------------------------------------

從 Then 後面的式子就看不懂了。

另外，

             ---
為什麼它是取 lim  而不是取 lim  ，機率收斂的定義不是要用 lim   嗎？
             n→∞         n→∞                         n→∞


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