題組內容

3. Let $X$ and $Y$ be random variables and $\{X_n, n \ge 1\}$ be a sequence of random variables. Define

$$\rho(X) = E\left(\frac{|X|}{1+|X|}\right).$$

Show that

(c) $\lim{n \to \infty} \rho(Xn - X) = 0$ if and only if $X_n \to X$ in probability. (15%)