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Let $X_n$ be a sequence of random variables such that $X_n>0$ a.s. for all $n$ and such that $X_n\stackrel{p}{\rightarrow} X$ with $X>0$ a.s.. I want to show that $\frac{1}{X_n}\stackrel{p}{\rightarrow}\frac{1}{X}$.

I found a similar topic here Convergence in probability inverse of random variable but it is restricted to the case $X=1$ and I need the most general one.

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