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Let $X_n\xrightarrow{p}0$ and $Z_n\xrightarrow{d}Z$ with $\mathbb{E}(|Z|)<+\infty$ and $Z$ is non-degenerate. I want to show that the sequence $\{X_nZ_n\}$ converges to $0$ in probability using only the definition of convergence in probability. I was thinking about the following: $\mathbb{P}(|X_nZ_n|>\varepsilon)\leq \mathbb{P}(|X_n|>\sqrt{\varepsilon})+\mathbb{P}(|Z_n|>\sqrt{\varepsilon})$ and then using Chebyshev's inequality at some point, but then I am not sure how to relate this to the convergence in distribution of $Z_n$ to $Z$.

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Let $\epsilon >0$. Choose $N$ so large that $P\{|Z| >N\} <\epsilon$ and $\pm N$ are continuity points of $F_Z$. Since $P\{|Zn| >N\} \to P\{|Z| >N\}$ we get $P\{|Zn| >N\}<\epsilon $ for $n$ sufficiently large. Now $P\{X_nZ_n| > \epsilon\} \leq P\{|X_n| >\epsilon /N\} + P\{ |Z_n| >N\}<P\{|X_n| >\epsilon /N\}+\epsilon$ for $n$ sufficiently large. Can you take it from here?

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