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Say $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ are two sequences of random variables defined on the same space such that $$W_n:= X_n - Y_n$$ converges in probability to zero.

Does it hold that $$Z_n:= \biggr \rvert \frac{1}{X_n} - \frac{1}{Y_n}\biggr \rvert$$ also converges in probability to zero?

(Unfortunately, I cannot assume that $\frac{1}{X_n Y_n}$ is $O_P(1)$, which would make things much easier.)

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$X_n = \frac1n$ and $Y_n = \frac1{n^2}$ provides a counterexample which is not even that much of a random sequence

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