shows that: $X_n \to 0$ in probability if and only if $\lim_{n \to \infty} E[|X_n|/(1+|X_n|)]=0$

Asked Jun 08 '23 at 19:13

Active Jun 08 '23 at 19:23

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For the first leg I know that by definition of convergence in probability I have to $\lim_{n \to \infty }P(|X_n|>\epsilon) =0$ And I think I can use that $Y_n = |X_n|/(1+|X_n|)$ But then I don't know how to proceed Thanks, Leo

edited Jun 08 '23 at 19:23

asked Jun 08 '23 at 19:13

Leonardo Martinez

Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – CrSb0001 Jun 08 '23 at 19:20
This has been answered on this site before – FShrike Jun 08 '23 at 19:23

shows that: $X_n \to 0$ in probability if and only if $\lim_{n \to \infty} E[|X_n|/(1+|X_n|)]=0$

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