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Suppose, for each $n$, $X_n$ is a random variable over $\{0, 1/n,2/n,...,1 \}$ and consider a sequence of random variables $\{X_n \}$. Then, can we construct a subsequence $\{n'\}$ such that, for each $x \in [0,1]$, we have $Pr (X_n' =x) \rightarrow Pr (X=x) $ as $ n' \rightarrow \infty$?

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    What is $X$? How is it distributed? Remember that if $x$ is irrational, $P(X_{n'}=x)=0$ for any sub sequence $\{n'\}$.2011-12-26

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