Suppose that $|X_n - Y_n|$ converges in probability to 0, and that $X_n$ converges in distribution to X. Show that $Y_n$ converges in distribution to X.
Thanks in advance.
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Goal: show $P(Y_n \le t) \to P(X \le t)$ [for $t$ at which $F_X(t):=P(X \le t)$ is continuous].