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$X_n$ is a sequence of random variables.$X_n \equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n \to \lim_{n\to\infty} a_n$ in probability.

I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.

kris91
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    write down the definition of convergence in probability and see what you can do – mm-aops May 30 '14 at 11:45
  • Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent? – DanZimm May 30 '14 at 11:55
  • no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence. – kris91 May 30 '14 at 14:07

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For any real number $a$ and $\varepsilon > 0$, $$ \Pr\{\left\lvert X_n-a\right\rvert\gt \varepsilon \}= \begin{cases} 1&\mbox{ if }\left\lvert a_n-a\right\rvert\gt \varepsilon;\\ 0&\mbox{ if }\left\lvert a_n-a\right\rvert\leq \varepsilon. \end{cases} $$

Davide Giraudo
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