P P P
Suppose X_n ----> X and Y_n ----> Y. Then X_n + Y_n ----> X + Y.
Proof:
Letε＞0 be given. Using the triangle inequality, we can write
│X_n - X│+ │Y_n - Y│≧│(X_n + Y_n) - (X + Y)│≧ε.
Since P is monotone relative to set containment, we have
P[∣(X_n + Y_n) - (X + Y)│≧ε]
≦ P[│X_n - X│+ │Y_n - Y│≧ε]
≦ P[│X_n - X│≧ε/2] + P[│Y_n - Y│≧ε/2]
By the hypothesis of the thoerem, the last two terms converge to 0
which gives us the desired result.
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請問Since P is monotone relative to set containment是什麼意思？
又，我不瞭解黃色不等式為什麼成立。(我猜測可能是用集合的範圍。)
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