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Suppose that $X_n, Y_n$ are sequences of random variable on probability space $\Omega$. If $Xn,Yn$ converges to $X$ ( some random variable ) in distribution, then is $X_n=Y_n$ almost everywhere (a.s)?

I supposed it is true, and I tried to prove $P\{|X_n-Y_n|>\epsilon\}=0$. I started from adding and subtracting $X$.

Am I in the right path ?

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The assertion is not true. A counterexample: Pick a random variable $Y$ and define $Y_n:=Y$ and $X_n:=Y+1/n$. Then $X_n$ converges in distribution to $Y$, and obviously $Y_n$ converges in distribution to $Y$. But $X_n$ and $Y_n$ are equal nowhere.

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