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If $X_n$ converges to $X$ almost surely how can we prove that $\cfrac 1{X_n}$ will converge to $\cfrac 1{X}$ almost surely?

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The limit of the quotient of two functions is the quotient of their limits if the limit in the denominator is not equal to 0. –  wnvl Oct 25 '12 at 21:31

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You might want to check if this result is true. Since $g(t)=1/t$ is continuous a.e., $g(X_{n})$ converges to $g(X)$ a.s.

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