A question about convergence with probability one

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Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this?

Thanks a lot.
 

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disregardthat
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Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this?

Thanks a lot.
I don't know what context this is in, but my answer would be that (Xn,Yn) converges to (X,Y) is equivalent to the statement that Xn converges to X and Yn converges to Y (with respect to any topology). Hence, if we treat these events as A and B respectively, you know P(A) = 1, P(B) = 1, hence [tex]P(A \cap B) = P(A)+P(B) -P(A \cup B) =1[/tex].
 

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