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A sequence of random variables $\{X_n\}$ converges to $X$ in probability if for any $\varepsilon > 0$, $$P(|X_n-X| \geq \varepsilon) \rightarrow 0$$

They converge in distribution if $$F_{X_n} \rightarrow F_X$$ at points where $F_X$ is continuous.

(There is another equivalent definition of converge in distribution in terms of weak convergence.)

It seems like a very simple result, but I cannot think of a clever proof.

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    Have you tried the wikipedia article: http://en.wikipedia.org/wiki/Proofs_of_convergence_of_random_variables#propA2 ? Most books on probability theory include a proof.2012-11-14
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    Oh, how come I didn't find it! It looks like something I have in mind. Thank you so much!2012-11-14

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