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m Reverted edits by 103.27.9.50 (talk) to last version by 98.255.224.120 |
GoingBatty (talk | contribs) m -, typo(s) fixed: Therefore → Therefore, (2) |
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</math>
On the right-hand side, the first term converges to zero as ''n'' → ∞ for any fixed ''δ'', by the definition of convergence in probability of the sequence {''X<sub>n</sub>''}. The second term converges to zero as ''δ'' → 0, since the set ''B<sub>δ</sub>'' shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem. Therefore, the conclusion is that
: <math>
\lim_{n\to\infty}\Pr \big(\big|g(X_n)-g(X)\big|>\varepsilon\big) = 0,
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\lim_{n\to\infty}X_n(\omega) = X(\omega) \quad\Rightarrow\quad \lim_{n\to\infty}g(X_n(\omega)) = g(X(\omega))
</math>
at each point ''X''(''ω'') where ''g''(·) is continuous. Therefore,
: <math>\begin{align}
\operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X)\Big)
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